What's a mathematician to do? (2010)(mathoverflow.net) |
What's a mathematician to do? (2010)(mathoverflow.net) |
> 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.
Thank you for highlighting that answer. It is one of my favourite pieces of writing about the culture of mathematics. I just want to add that that particular answer is now affectionately known as Thurston's Paean.
For example. Take a modern country with a modern economy. Flatten it. Destroy all the factories. Bankrupt all the companies. You can get back to a fully modern economy again quite quickly. WWII demonstrates it.
Taking an unindustrialized country through the development process... that's very tricky. It can't really be rushed.
For a long time, economic development was seen as mostly capital and technology. You need time to develop all the capital needed. Roads, factories, etc. But... development efforts underperformed. Then the idea of "human capital" got popular as a way of explaining the deficit. Education, mostly. Development efforts still underperformed.
I think the "living community" thing is the answer to this. It' ecology. You can't make a rainforest by just dumping all the necessary organisms into the right climate. It's the endlessly complicated relationships between all those organisms that make the rainforest.
This is one of the things that worries me about the pace of modern change. When writing and literacy resurged in classical antiquity... we totally lost all the ways of (for example) doing scholarship orally. Socrates (through Plato) wrote about some of the downsides to this.
...and we did completely lose oral scholarship. We have no idea how to do it. Once the living culture died... it stayed dead. All the knowledge contained within it went away.
I agree. A body of knowledge, mathematical knowledge being one of them, is a give-and-take between its producers and consumers; a market for ideas. It grows in that ecology of people with its pathfinders, specialists, generalists, historians, educators, etc. Committing to a body of knowledge is becoming part of its living culture.
Where I disagree: I believe some of the loss is inevitable. Keeping in mind the example of a body of knowledge above, as the scale of what's accumulated until now grows, the role each of us play in the sustenance of its culture shrink. This is a direct consequence of the modern developmental process (ie division of labour to the point where it feels like we are all modular parts of a much larger whole).
I can't say whether its better to focus on recovering what's lost, or, trust the process, as it were.
Theres a limited amount of time, space and energy so what's the ideal mechanism to say what to pay attention to or not
It would appear that LLMs are invalidating this claim. Things can live in synthetic form and carry on just fine. Instead of cultivating a population of learned minds we are just feeding a few dozen egregores of models and training corpuses.
LLMs are quite good at simulating life and living intelligence (in the short term), but they aren't any of that. That's why we call it artificial intelligence. It's true that we can't put our finger on what exactly the difference is, but it's not like reality has ever felt encumbered by our limited understanding.
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.
Far from being motivated by some applications, the most useful discoveries in mathematics are usually discovered "for their own sake" and their application is only discovered later. Sometimes centuries later!
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.
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.
I had a conversation a day ago with a couple of high-school students who, were obviously smart (and a bit on the spectrum), but also lacking in broad knowledge, as one would expect from high-school students.
I think it revealed something about that sense of 'magic' or inaccessible talent. One of them mentioned the fast inverse square root function and marvelled about how even anyone could even come up with an idea like that because it seemed to him to be some transcendent feat to be able to realise casting binary floats into ints would be useful. They had looked at the function and couldn't fathom how something like that worked.
We had no computer to hand, but I asked him what 10^100 divided by 10^10 was. Smart kid that he was, answered instantly, correctly. I then asked him what operation he performed in his head to do that, and noted floating point exponents are just using 2 instead of 10. On the spot he figured out how the fast inverse square root worked. The magic vanished and it became accessible. Some people lament the loss of 'magic' in this way, but I think the thing that makes it special, in this instance and in the universe in general, is that it still works, and it isn't magic. It's real and the fact that it be that without invoking some unaccessible property makes it even more special
On the contrary, prompting LLMs creates a whole lot of newly accessible basic work.
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.
"That view is that there is still a great deal of value in struggling with a mathematics problem, but that the era where you could enjoy the thrill of having your name forever associated with a particular theorem or definition may well be close to its end. So if your aim in doing mathematics is to achieve some kind of immortality, so to speak, then you should understand that that won’t necessarily be possible for much longer — not just for you, but for anybody."
He may seem to imply the end is only for some subset of reasons but if you read the entire essay he is just trying to give hope where the rest of the essay is really damning!
... they will discover that level is already crowded by LLMs
It must feel similar to those who wanted to become chess or go masters after computers surpassed humanity in those games.
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.
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...
When I was in grad school, we learned about wavelets, but we did research on convex optimization for statistics. The first was an accomplishment of the last generation of mathematicians, and it would be hard to publish something groundbreaking. But nobody had really considered sparsity inducing optimization, so that was our problem.
In many ways, the situation is somewhat better for applied mathematicians because the problem space is wider. Ingrid Daubechies was an applied mathematician, and her work on wavelets was an outgrowth of work that originated in the petroleum exploration community. Wild how these connections get made.
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 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.
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.
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)
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
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.
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.
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.
Like prime numbers? (used in cryptography)
On the contrary, complex numbers were introduced to make the cubic formula work.
Hard to imagine now, but back when he started out, there were really no (to very few!) accessible math tutoring vids on the video platforms. Most of the times you had some universities, like MIT, putting out long-form vids from lectures - but actually having easily digestible 5 min vids like those Khan put out, just wasn't a thing.
I am not a mathematician, but out of personal interest, I have worked with professional mathematicians in the past to help refine notes that explain certain intermediate steps in textbooks (for example, Galois Theory, by Stewart, in a specific case). I was surprised to find that it was not just me who found the intermediate steps of certain proofs obscure. Even professional mathematicians who had studied the subject for much of their lives found them obscure. It took us two days of working together to untangle a complicated argument and present it in a way that satisfied three properties: (1) correctness, (2) completeness, and (3) accessibility to a reasonably motivated student.
And I don't mean that the books merely omit basic results from elementary topics like group theory or field theory, which students typically learn in their undergraduate courses. Even if we take all the elementary results from undergraduate courses for granted, the proofs presented in graduate-level textbooks are often nowhere near a complete explanation of why the arguments work. They are high-level outlines at best. I find this hugely problematic, especially because students often learn a topic under difficult deadlines. If the exposition does not include sufficient detail, some students might never learn exactly why the proof works, because not everyone has the time to work out a 10-page proof for every 10 lines in the book.
Many good universities provide accompanying notes that expand the difficult arguments by giving rigorous proofs and adding commentary to aid intuition. I think that is a great practice. I have studied several graduate-level textbooks in the last few years and while these textbooks are a boon to the world, because textbooks that expose the subject are better than no textbooks at all, I am also disappointed by how inaccessible such material often is. If I had unlimited time, I would write accompaniments to those textbooks that provide a detailed exposition of all the arguments. But of course, I don't have unlimited time. Even so, I am thinking of at least making a start by writing accompaniment notes for some topics whose exposition quality I feel strongly about, such as s-arc transitivity of graphs, field extensions and so on.
That sounds dramatic, but it’s really obvious if you think about it. Right now, a person has to study for about 20 years (on average) to make novel contributions in mathematics. They have to learn what’s come before, the techniques, the results, etc. If mathematics continues, eventually it could take 25 years, or 30 years, or even a whole lifetime. At some point, most people will not be able to understand the work that’s been done in any subfield (or the work required to understand a subfield) in a human’s life. I claim this is the logical end of mathematics, at least as a human endeavor.
Now, there will be some results which refine other work and simplify results, but being able to teach a rapidly growing body of literature efficiently will be important to stave off the end of mathematics.
To your point, I think you're right. I'm not in mathematica, but the value of good pedagogy on shrinking the time it takes to get people to the forefront of any field feels like it's heavily overlooked.
https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
This is somewhat misleading, the LLMs' contributions are in a limited niche of highly technical problem solving. They're neat but they're not the first mathematical theorem that gets automatically solved by a computer, that was done already in the 1990s.
> Maybe by using LLMs as a mighty tool and providing skilled usage and oversight?
Yes, even in the areas where LLMs are at their best, we'll still need a lot of human effort to make the results cleanly understandable. LLMs cannot do this well, even their generated papers have to be rewritten by human experts to surface the important bits.
The can't predict the consequences of an action predicting one token after another. They can't solve a Rubik's Cube unlike a 7 year old human who can learn to do it in a weekend. They can't imagine the perspective of being a human being unlike a 7 year old human if asked to imagine they where in the position of another human.
Can you imagine what if feels like to be a LLM?
Can one LLM have a better sensation of what it feels like to be a different LLM (say one that score a little better?)?
You design circularly defined criteria...
On the other hand, in the very long run, what does it mean if a talented human being does not have enough years of life to fully analyze and understand an extremely advanced proof created by AI?
[0]: https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
The most recent advances are stunts by a handful of famous prompters who are funded in various ways by the LLM industrial complex.
How many theorems are proven by mathematicians each year? Let's guess 10000. Then the Erdos toy proofs with unknown token and resource usage are less than 1%.
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)
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.
This is not the fault of the mathematicians.
Ironically, there is a shoe company pivoting to AI. My taxi driver told me buy the stock:
or even just take a picture of the thing, since they can digest visual input now
It's education for whoever finds it educational
Plumbing, react, combinatorics, real analysis, python, c++, cad, micro and macro economics, reinforcement learning, to name just a few of the things I learned through YouTube.
We don’t give enough credit to what we take for granted today.
This is exactly what Numberphile does. Those who are hooked will find the resources on their own. They need a reason to look for them and Numberphile gives them one.
However I will push back on the claim that inspiration is not education. It may not be sufficient on its own when resources are not readily available. But now that they are it's the inspiration and persistence that are the missing magic sauces.
Young people are curious, sometimes all they need is a spark and to be introduced to a new topic in an engaging way. These forms of content deliver that spark.