After reading another post about the most recent advances LLMs have made in finding and writing up novel, correct proofs, it sounds like the frontier models are now at the point of PhD student level. I wonder how a math student could contribute today, if they're just starting on the PhD track? Maybe by using LLMs as a mighty tool and providing skilled usage and oversight?
It must feel similar to those who wanted to become chess or go masters after computers surpassed humanity in those games.
If we see our contributions as brownian motion rather than preconceived trajectories, then, rather than focusing on the Gausses, Einsteins, Patons as providing singular progress, they become the the dominant least energy paths to what we recognize as truth. Without negating the individual’s contribution, the ones we see as truly important are the ones that supported by every other’s attempt, finds the path forward.
This should provide hope, if we can leave aside our egos and focus on humanity, we can, and do, all contribute even though a few seems to get all the credit.
This also goes for AI, it may be an accelerant in research, but the probability distribution of reality is large, large enough for humans to wonder, ask questions and stumble upon a new path forward, that computers alone don’t find.
> mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification.
Yes! And this applies to all human culture, not just math. Everything people have figured out needs to be in living form to carried on. The more people the better. If math, or any product of human skill, is only recorded in papers or videos, that isn't the same as having millions of people understanding it in their own ways.
Modern culture often emphasizes innovation and fails to value mere maintenance, tradition, and upkeep. This can lead to people like the OP feeling that they have nothing to contribute, when actually, just learning math, being able to do it, being able to help others learn it - all of these are contributions.
We are all needed to keep civilization afloat, in ways we cannot anticipate. We all need to pursue some kind of excellence just to keep human culture alive.
Do the math because you enjoy doing the math and if you do it long enough you may well do something of value to someone else. Same goes for most intellectual and artistic pursuits I think.
I’ve learned for myself that as soon as enjoyment is based on some future achievement or ranking my work against others the day to day satisfaction dries up.
So I've got a gut feeling that math (like human languages (like programming languages)) is best learned in service of some greater end.
I look at some truly impressive projects like CLASP which sprang into existence not because of someone noodling around, but because they had a bigger goal which required the team build it.
So my advice to any mathematician who feels lost, like they don't know what to work on, would be to go collaborate with someone who has an actual goal, to look for inspiration in the kinds of math they need.
Today, there are a lot of opportunities to jump forward that only get capitalized on through coincidence (e.g. two people bump into each other at a conference, or researcher happens to have a colleague working on a related problem through the lens of a different discipline). If AI does nothing but guarantee that everyone will have such a coincidence by serving as that expert from a different discipline, that will still be a massive driving force for progress.
The question of "whats a mathematician to do" is still clear: you need to find and curate and clearly express interesting and valuable problems.
I think it is like for a programmer to ask "How can one contribute to computer science?", while thinking about people like, Dijkstra, Knuth, or maybe even Carmack.
There are some geniuses who do groundbreaking work, but this wouldn't be of much use it it wasn't for the millions of people who do actual work with these theories (applied math), and teachers who train the next generation. In the academia, small discoveries exist too, these can be the stepping stones for the big things to come, even if they don't have a direct application now.
Today, even understanding what new mathematics is being done in a particular zone of the mathematical universe appears to require a four-year graduate program of constant study just to be able to follow some mathematician’s original work - and that’s only going to give you a window into a rather narrow subsection of mathematics. The days when people of great talent like Euler and Gauss could contribute to many areas of mathematics are long gone.
But mere mortals can still derive great satisfaction from following along in the footsteps of past pioneers, possibly adapting their work to new problems in a minor way, or just creating educational visualizations and tools that help other people understand things like Galois theory, Poincare phase space or Markov chains, which can be applied to quantum mechanics, orbital dynamics, or protein sequence analysis. That’s valuable, even if no Fields Medals will be coming your way.
For the core discipline, though, I’d mostly worry about lack of opportunities for serious mathematicians to practice their craft in the USA due to the trends of academic budget cuts, anti-intellectual rhetoric, insistence on profit generation as the only rationale for doing anything, etc. Looks a bit 1930s Germany to me, at least here in the USA.
a) Individualized teaching methodology. We come with different backgrounds, therefore different types of analogies/examples, different levels of background material, different (but systematized) levels of presentation should be used. The same ask should be applied to kids learning through starting at preschool.
b) Readable mathematics papers where the compact notations are abandoned, and narrative, visualizations are introduced, while preciseness is maintained. It is possible that the same paper (or chapter or topic) should be renderable in multiple ways (for professional mathematicians in the field, for a casual reader, for a student, for an individual reader (as for (a) )
c) Mathematical logic / tooling for differentiable data/event computing. Where there are mathematical tools as well as CS implementation of this tools that allow to act on a difference in state, data, actions.
Typical mathematics (with exception of may be time series), does not view time as 'first class citizen' so to speak, be it abstract algebra and category theory or something else. But, I think, when we go to the 'applied world' we must introduce 'time dimension' as first class citizen. So having the mathematical machinery dealing with this dimension in organic way across many of the areas of mathematics -- will be beneficial to the application of this one of the most valuable human tools.
I've been thinking a lot about your point b) over the years.
I'm conceptualizing a piece of knowledge as an interface that can be `implemented` but with different classes (explanations renderings for different audiences).
For example, the "derivative interface" represents knowledge of the concept of derivative operations and basic skills to compute derivatives of various functions. The interface doesn't specify HOW to teach this topic or HOW DEEP, so there are multiple implementations:
- basic visual explanations (for kids)
- basic algebra steps (for high school)
- standard explanation (for undergraduate students)
- compact explanation (a reviee for grad students)
The above implementation are polymorphism due to the "reader level of knowledge," but there could be other, e.g. derivatives explained using code like in Sec 4.1 in this calculus tutorial[1].
It would be A LOT of work to produce all these explanations but it would make for a kick ass math textbook that you can pick up and learn, no matter what your level is (instead of getting lost or bored and looking for another resource).
@ivansavz, thank you for the followup and sharing your thoughts on b).
Some time ago, when my Dad asked me to teach him a bit of programming, I made a huge mistake. I was arrogant, and thinking to teach him in a way I learned.
And it was totally wrong, totally wrong on many levels on the approach, on the emotional aspect of it.. just totally wrong.
He is no longe with me, and I keep coming back to this, as I cannot fix it.
So from that time, gradually I started unpacking, if he were alive today -- how would I do that differently.
It is at that time (now 20+ years ago) that I started coming up with these
personalized learning, audience specific rendering of the material. And think these two need to be combined together.
My Dad had different analogies and reference points than me, and also he was brighter, faster in many ways, while I tend to be slower and less visual and I have easier time with hypotheticals/and abstractions.
So the personalized part has to reflect the differences. Another example, every time I open a book on statistics I see pocker, cards and various other things -- I have no idea about. These are not great analogies/examples for me as I struggle to grasp the context.
Yet, I also appreciate (now, when it is late) that others are different than me.
So my thought here in a way, similar to yours but merging together student's-level plus students personal experience/background.
So in that context I would say
1). Each student has to built out (may be even gradually) a profile of preferences (background, subject level, visualization proficiency, many other nuanced cognitive differences). May be bulding it answering some sort of logntitudional survey (over time) is a right approach here.
2) The presentation material would be separate into
core , presentation, assessment
3) core stays the same and developed by the core instructor/teacher author
4) presentation is developed my multiple means and authors
4i) by the author(s) themselves adopting some 'default student profiles)
4ii) by author(s) authorised contributors that develop materias for other student profiles
5) assessments done in same way as (4).
Then when I as student order (buy) or download or subscribe to the given textbook or a class
I specify my profile (that's built in (1)), may be some other learning preference, click 'generate' (or subscribe if that's an online class)
and I get the 'rendered' material that I then use.
(of course if the whole system also has online presense, then there is a benefit of a 'forum-like' community around similary-rendered materials -- as it will have folks with, presumably, similar profiles, and the same about assessments).
--
I agree with you that the students agent may dictate the type of rendering so to speak for the materials. In way, i am thinking to capture that with the longtitudional survey updating student profile, through their lifelong learning journey.
WRT a lot of work, agreed I am hoping that 4ii) is an attempt to partially address it.
> One can rewrite their books in modern language and notation or guide others to learn it too but I never believed this was the significant part of a mathematician work
There's yer problem right there. Good pedagogy is hard and highly undervalued. IMHO Grant Sanderson (a.k.a. 3blue1brown) is making some of the most significant contributions to math in all of human history by making very complex topics accessible to ordinary mortals. In so doing he addresses one of the most significant problems facing humankind: the growing gap between the technologically savvy and everyone else. That gap is the underlying cause of some very serious problems.
I think the answer is to do multi-disciplinary work.
Venture outside of pure theoretical math. Learn some other domain knowledge and combine it with your mathematical ommph. That's the easiest way to make an impact now rather than potentially decades later.
To add onto these answers, there were some notable "outsider art" additions to math around this time.
A few months before this post, Futurama contributed a new proof to the mathematical canon (for "The Prisoner of Benda"), resolving the conflict of the episode.
Almost a year after posting this, a 4chan user solved a previously-unsolved superpermutation (combinatorics) problem in a discussion about anime.
I think everyone who has thought about math seriously has felt similarly to the OP. It was impressed upon me early on that there are combinatorically (hah) many combinatorics problems to be solved and that these were just a few.
this guy is resigned to feel worthless compared to other mathematicians (suggesting he become cannon fodder in some type of mathematical sacrifice. i wonder if that analogy even makes sense in the field xD).
but, he desperately wants to become a great mathematician who creates completely original work.
from my experience, people tend to or even want to limit themselves. they think they know the ceiling of their capabilities and it becomes some self fulfilling prophecy.
if you really care about doing something great like this guy does, don't limit yourself. push until you achieve the greatness you want to achieve.
it's like that one saying, aim for the stars and you might land on a cloud. you will be surprised at how capable you actually are
I think we also have to be honest and admit that, yes, indeed, there is less novel maths for all of us to be doing. The pioneers came first and discovered a lot of low hanging fruit. There were a lot of geniuses that mined the rest and reached higher in the tree. Now even the smartest mathematicians are left solving abstract puzzles with little utility in the real world. (Don't get me wrong, it's very fun, and sometimes useful too.)
After my PhD in applied mathematics, I decided to leave the field, partly because I feel it really has advanced so far that new discoveries do little to move the needle in the real world. There's enough smart people who obsess over nothing else but maths that I can go and do more practical stuff...
I am glad to see this today - after reading Tim gower's recent post on chatgpt 5.5 pro's phd level research ability I was feeling slightly sad about the future of math research.
Interestingly enough, the moment I saw the title I thought of Bill Thurston's famous article "On proof and progress in mathematics" and the top comment on the OP's thread is from him! Reading his reply sort of gave me the antidote to the temporary blues I felt yesterday.
31 comments
[ 1.5 ms ] story [ 55.3 ms ] threadIt must feel similar to those who wanted to become chess or go masters after computers surpassed humanity in those games.
This also goes for AI, it may be an accelerant in research, but the probability distribution of reality is large, large enough for humans to wonder, ask questions and stumble upon a new path forward, that computers alone don’t find.
> mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification.
Yes! And this applies to all human culture, not just math. Everything people have figured out needs to be in living form to carried on. The more people the better. If math, or any product of human skill, is only recorded in papers or videos, that isn't the same as having millions of people understanding it in their own ways.
Modern culture often emphasizes innovation and fails to value mere maintenance, tradition, and upkeep. This can lead to people like the OP feeling that they have nothing to contribute, when actually, just learning math, being able to do it, being able to help others learn it - all of these are contributions.
We are all needed to keep civilization afloat, in ways we cannot anticipate. We all need to pursue some kind of excellence just to keep human culture alive.
But unfortunately human knowledge accumulation and advancement over the last many thousand years has been pretty large deep and varied.
Finding something novel for phds or profits or crime or whatever th fk is harder everyday.
Chaining unrelated sentences is retarded. Chaining sentences like most people is common sense. Chaining sentences airtight is math.
You ask what a true mathematician does. He chains sentences like everyone else but with an effort to make them airtight.
Do the math because you enjoy doing the math and if you do it long enough you may well do something of value to someone else. Same goes for most intellectual and artistic pursuits I think.
I’ve learned for myself that as soon as enjoyment is based on some future achievement or ranking my work against others the day to day satisfaction dries up.
I look at some truly impressive projects like CLASP which sprang into existence not because of someone noodling around, but because they had a bigger goal which required the team build it.
So my advice to any mathematician who feels lost, like they don't know what to work on, would be to go collaborate with someone who has an actual goal, to look for inspiration in the kinds of math they need.
Today, there are a lot of opportunities to jump forward that only get capitalized on through coincidence (e.g. two people bump into each other at a conference, or researcher happens to have a colleague working on a related problem through the lens of a different discipline). If AI does nothing but guarantee that everyone will have such a coincidence by serving as that expert from a different discipline, that will still be a massive driving force for progress.
The question of "whats a mathematician to do" is still clear: you need to find and curate and clearly express interesting and valuable problems.
There are some geniuses who do groundbreaking work, but this wouldn't be of much use it it wasn't for the millions of people who do actual work with these theories (applied math), and teachers who train the next generation. In the academia, small discoveries exist too, these can be the stepping stones for the big things to come, even if they don't have a direct application now.
But mere mortals can still derive great satisfaction from following along in the footsteps of past pioneers, possibly adapting their work to new problems in a minor way, or just creating educational visualizations and tools that help other people understand things like Galois theory, Poincare phase space or Markov chains, which can be applied to quantum mechanics, orbital dynamics, or protein sequence analysis. That’s valuable, even if no Fields Medals will be coming your way.
For the core discipline, though, I’d mostly worry about lack of opportunities for serious mathematicians to practice their craft in the USA due to the trends of academic budget cuts, anti-intellectual rhetoric, insistence on profit generation as the only rationale for doing anything, etc. Looks a bit 1930s Germany to me, at least here in the USA.
b) Readable mathematics papers where the compact notations are abandoned, and narrative, visualizations are introduced, while preciseness is maintained. It is possible that the same paper (or chapter or topic) should be renderable in multiple ways (for professional mathematicians in the field, for a casual reader, for a student, for an individual reader (as for (a) )
c) Mathematical logic / tooling for differentiable data/event computing. Where there are mathematical tools as well as CS implementation of this tools that allow to act on a difference in state, data, actions.
Typical mathematics (with exception of may be time series), does not view time as 'first class citizen' so to speak, be it abstract algebra and category theory or something else. But, I think, when we go to the 'applied world' we must introduce 'time dimension' as first class citizen. So having the mathematical machinery dealing with this dimension in organic way across many of the areas of mathematics -- will be beneficial to the application of this one of the most valuable human tools.
I'm conceptualizing a piece of knowledge as an interface that can be `implemented` but with different classes (explanations renderings for different audiences).
For example, the "derivative interface" represents knowledge of the concept of derivative operations and basic skills to compute derivatives of various functions. The interface doesn't specify HOW to teach this topic or HOW DEEP, so there are multiple implementations:
The above implementation are polymorphism due to the "reader level of knowledge," but there could be other, e.g. derivatives explained using code like in Sec 4.1 in this calculus tutorial[1].It would be A LOT of work to produce all these explanations but it would make for a kick ass math textbook that you can pick up and learn, no matter what your level is (instead of getting lost or bored and looking for another resource).
[1] https://minireference.com/static/tutorials/calculus_tutorial...
Some time ago, when my Dad asked me to teach him a bit of programming, I made a huge mistake. I was arrogant, and thinking to teach him in a way I learned. And it was totally wrong, totally wrong on many levels on the approach, on the emotional aspect of it.. just totally wrong.
He is no longe with me, and I keep coming back to this, as I cannot fix it.
So from that time, gradually I started unpacking, if he were alive today -- how would I do that differently.
It is at that time (now 20+ years ago) that I started coming up with these
personalized learning, audience specific rendering of the material. And think these two need to be combined together.
My Dad had different analogies and reference points than me, and also he was brighter, faster in many ways, while I tend to be slower and less visual and I have easier time with hypotheticals/and abstractions.
So the personalized part has to reflect the differences. Another example, every time I open a book on statistics I see pocker, cards and various other things -- I have no idea about. These are not great analogies/examples for me as I struggle to grasp the context.
Yet, I also appreciate (now, when it is late) that others are different than me.
So my thought here in a way, similar to yours but merging together student's-level plus students personal experience/background.
So in that context I would say
1). Each student has to built out (may be even gradually) a profile of preferences (background, subject level, visualization proficiency, many other nuanced cognitive differences). May be bulding it answering some sort of logntitudional survey (over time) is a right approach here.
2) The presentation material would be separate into core , presentation, assessment
3) core stays the same and developed by the core instructor/teacher author
4) presentation is developed my multiple means and authors 4i) by the author(s) themselves adopting some 'default student profiles) 4ii) by author(s) authorised contributors that develop materias for other student profiles
5) assessments done in same way as (4).
Then when I as student order (buy) or download or subscribe to the given textbook or a class I specify my profile (that's built in (1)), may be some other learning preference, click 'generate' (or subscribe if that's an online class) and I get the 'rendered' material that I then use.
(of course if the whole system also has online presense, then there is a benefit of a 'forum-like' community around similary-rendered materials -- as it will have folks with, presumably, similar profiles, and the same about assessments).
--
I agree with you that the students agent may dictate the type of rendering so to speak for the materials. In way, i am thinking to capture that with the longtitudional survey updating student profile, through their lifelong learning journey.
WRT a lot of work, agreed I am hoping that 4ii) is an attempt to partially address it.
There's yer problem right there. Good pedagogy is hard and highly undervalued. IMHO Grant Sanderson (a.k.a. 3blue1brown) is making some of the most significant contributions to math in all of human history by making very complex topics accessible to ordinary mortals. In so doing he addresses one of the most significant problems facing humankind: the growing gap between the technologically savvy and everyone else. That gap is the underlying cause of some very serious problems.
Venture outside of pure theoretical math. Learn some other domain knowledge and combine it with your mathematical ommph. That's the easiest way to make an impact now rather than potentially decades later.
A few months before this post, Futurama contributed a new proof to the mathematical canon (for "The Prisoner of Benda"), resolving the conflict of the episode.
Almost a year after posting this, a 4chan user solved a previously-unsolved superpermutation (combinatorics) problem in a discussion about anime.
I think everyone who has thought about math seriously has felt similarly to the OP. It was impressed upon me early on that there are combinatorically (hah) many combinatorics problems to be solved and that these were just a few.
but, he desperately wants to become a great mathematician who creates completely original work.
from my experience, people tend to or even want to limit themselves. they think they know the ceiling of their capabilities and it becomes some self fulfilling prophecy.
if you really care about doing something great like this guy does, don't limit yourself. push until you achieve the greatness you want to achieve.
it's like that one saying, aim for the stars and you might land on a cloud. you will be surprised at how capable you actually are
Bill Thurston's answer to “What's a mathematician to do?” (2010) - https://news.ycombinator.com/item?id=23461983 - June 2020 (21 comments)
Bill Thurston answers: What's a mathematician to do? - https://news.ycombinator.com/item?id=15578866 - Oct 2017 (25 comments)
What's a Mathematician to do? - https://news.ycombinator.com/item?id=8265509 - Sept 2014 (44 comments)
Bill Thurston's answer to "What's a mathematician to do?" - https://news.ycombinator.com/item?id=4419859 - Aug 2012 (1 comment)
Edit: bonus relateds:
https://news.ycombinator.com/item?id=43345503 (March 2025)
It's not mathematics that you need to contribute to (2010) - https://news.ycombinator.com/item?id=36744690 - July 2023 (65 comments)
Knots to Narnia – Bill Thurston (1992) [video] - https://news.ycombinator.com/item?id=34426275 - Jan 2023 (8 comments)
On Proof and Progress in Mathematics (1994) - https://news.ycombinator.com/item?id=31960487 - July 2022 (1 comment)
On Proof and Progress in Mathematics (1994) [pdf] - https://news.ycombinator.com/item?id=12280139 - Aug 2016 (8 comments)
Bill Thurston has died - https://news.ycombinator.com/item?id=4419566 - Aug 2012 (18 comments)
On Proof And Progress In Mathematics (1994) [pdf] - https://news.ycombinator.com/item?id=2582730 - May 2011 (1 comment)
On proof and progress in mathematics (1994) - https://news.ycombinator.com/item?id=982335 - Dec 2009 (5 comments)
After my PhD in applied mathematics, I decided to leave the field, partly because I feel it really has advanced so far that new discoveries do little to move the needle in the real world. There's enough smart people who obsess over nothing else but maths that I can go and do more practical stuff...
Interestingly enough, the moment I saw the title I thought of Bill Thurston's famous article "On proof and progress in mathematics" and the top comment on the OP's thread is from him! Reading his reply sort of gave me the antidote to the temporary blues I felt yesterday.