7 ms·

What's a mathematician to do? (2010)

179 points by ipnon 4mo ago

Schlagbohrer 4mo ago
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.
- generic92034 4mo ago
  I wonder if AI is one means to overcome the natural limits of human knowledge aggregation [0]. 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/ https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
  - morkalork 4mo ago
    Perhaps it will become like those cathedrals that took centuries and many generations of humans to build.
    - TimorousBestie 4mo ago
      Mathematics as an aggregate already is that cathedral. It is grander and more beautiful than any earthly cathedral.
    - generic92034 4mo ago
      Yes, but you (as a human) can still understand the cathedral (the building). This is not guaranteed for advanced AI work in mathematics in the future. If so, are we/they are really still adding to human knowledge, at this stage?
- dataviz1000 4mo ago
  LLM models can only predict the next token. 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.
  - DoctorOetker 4mo ago
    Those are very strong claims, do you really believe an LLM can't be trained to solve Rubik's Cubes? 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...
    - r_lee 4mo ago
      honestly I'm pretty sure opus could solve a rubiks cube if you just gave it the layout if the sides and looped until it would solve it or even just take a picture of the thing, since they can digest visual input now
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- 4qshT 4mo ago
  The Mathoverflow question was asked 15 years ago. The top answer says that the human community part is very important and spreading knowledge an critical thinking is valuable. 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%.
  - sdenton4 4mo ago
    ...And in 1900, how many carriages were horseless?
    - ppll_12 4mo ago
      In 2026, how many people X-ray their feet at the shoe store or have watches with radium paint? Ironically, there is a shoe company pivoting to AI. My taxi driver told me buy the stock: https://www.bbc.com/news/articles/c98mrepzgj7o https://www.bbc.com/news/articles/c98mrepzgj7o
- photochemsyn 4mo ago
  If your motivation is being recognized as the best of the best, winning the competition, yes it’s probably a bleak world. But if you motivation is improving your own capabilities, with the metric being if you’re better know then you were last month, then it’s not a bleak world, there are many more tools available to help you learn and improve now then there were in the past.
- zozbot234 4mo ago
  > 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. 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.
  - DroneBetter 4mo ago
    > done already in the 1990s by human-written programs that iterated through the finite casework that human thought had reduced the theorems to (four-colour theorem, FLT, etc.), which recent developments (eg. LLMs autonomously resolving Erdős problems) seem meaningfully distinct from. > human effort to make the results cleanly understandable well, perhaps loops of "derive proof through reasoning in English, formalise in Lean, use AST size of formal proof as a metric to optimise (via an LLM-guided search), translate back into English" could improve this? a lot of resources are being spent to make frontier LLMs more resistant to hallucinations via Lean, perhaps cogency will increase as a byproduct.
lokimedes 4mo ago
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.
- groundzeros2015 4mo ago
  Yeah I don’t think invention or technological development is inevitable or random. It’s path dependent and colored by individuals and culture.
getnormality 4mo ago
From one of the answers: > 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.
- rdevilla 4mo ago
  > Everything people have figured out needs to be in living form to carried on. 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.
  - gosub100 4mo ago
    All LLMs do is launder other people's IP. So I don't think you invalidated any other claim.
  - pegasus 4mo ago
    They are not invalidating this claim, and cannot, unless we'd actually try it out for a few generations. Which we shouldn't and won't. 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.
ucla_rob 4mo ago
Fortunately doing something novel is one of the main things llms can't do. 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.
- konschubert 4mo ago
  Maybe they can: https://news.ycombinator.com/item?id=48071262 https://news.ycombinator.com/item?id=48071262
elendilm 4mo ago
At the very foundation, chaining sentences together is what we call logic. 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.
ljasdjasdfasdf 4mo ago
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jebarker 4mo ago
“Comparison is the thief of joy.” 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.
- r_lee 4mo ago
  and so how exactly are you supposed to provide for your family in this scenario?
  - codechicago277 4mo ago
    Could work in a patent office or something.
  - jebarker 4mo ago
    By having a job. If that job is the same as your intellectual/artistic pursuit then you have to balance the needs of satisfying your employer and what keeps your motivation going over the long term. All I’m saying is that worrying too much about future achievement or “great contributions” are a recipe for burnout and disappointment.
bloaf 4mo ago
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.
- nimonian 4mo ago
  It's a delightful counterintuition that your gut feeling is mostly wrong: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf https://webhomes.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf 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!
  - dennis_jeeves2 4mo ago
    >are usually discovered "for their own sake" Like prime numbers? (used in cryptography)
  - skybrian 4mo ago
    If so that seems like an opportunity for people who want to work on applied math? There’s a big backlog of techniques that so far have not been useful.
    - kcexn 4mo ago
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GuB-42 4mo ago
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.
- s5300 4mo ago
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- bee_rider 4mo ago
  I do think this has been less pressing of a question for programmers. For the longest time there was infinite work to do, no matter your depth—implementing business logic, making frameworks more general, making sure things fit into cache lines—that was accessible to us non-geniuses. Maybe in the LLM era this sort of t-crossing and i-dotting will go away.
  - zozbot234 4mo ago
    > Maybe in the LLM era this sort of t-crossing and i-dotting will go away. On the contrary, prompting LLMs creates a whole lot of newly accessible basic work.
- Lerc 4mo ago
  >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. 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
photochemsyn 4mo ago
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.
dunham 4mo ago
For context - the top answer was written by Bill Thurston, who was awarded a Fields Medal. (Kind of like a nobel prize for mathematics.)
vladislavp 4mo ago
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.
- ivansavz 4mo ago
  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). [1] https://minireference.com/static/tutorials/calculus_tutorial.pdf#page=9 https://minireference.com/static/tutorials/calculus_tutorial...
lisper 4mo ago
> 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.
- LPisGood 4mo ago
  Indeed, pedagogy is important to staving off the end of mathematics. 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.
  - pertique 4mo ago
    There's a Scott Alexandar story that plays with this exact topic: Ars Longa, Vita Brevis [1] 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/ https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
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wcfrobert 4mo ago
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.
cryo32 4mo ago
Mathematican here. Writing software because it pays better.
lynndotpy 4mo ago
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.
baddash 4mo ago
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
siva7 4mo ago
There are many practical jobs left for mathematicians. Time to discover what you like to do with your hands.
dang 4mo ago
Related. Others? Bill Thurston's answer to “What's a mathematician to do?” (2010) - https://news.ycombinator.com/item?id=23461983 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 https://news.ycombinator.com/item?id=15578866 - Oct 2017 (25 comments) What's a Mathematician to do? - https://news.ycombinator.com/item?id=8265509 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 https://news.ycombinator.com/item?id=4419859 - Aug 2012 (1 comment) Edit: bonus relateds: https://news.ycombinator.com/item?id=43345503 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 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 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 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 https://news.ycombinator.com/item?id=12280139 - Aug 2016 (8 comments) Bill Thurston has died - https://news.ycombinator.com/item?id=4419566 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 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 https://news.ycombinator.com/item?id=982335 - Dec 2009 (5 comments)
- susam 4mo ago
  It's not mathematics that you need to contribute to (2010) - https://news.ycombinator.com/item?id=36744690 https://news.ycombinator.com/item?id=36744690 - July 2023 (65 comments)
  - dang 4mo ago
    Inserted. Thanks!
ashley95 4mo ago
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...
- _alternator_ 4mo ago
  Is this not just a perspective issue? The fields we learn about in school are the ones that are in some sense mature, that have been cashed. 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.
jdw64 4mo ago
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beyonddream 4mo ago
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.
- fasterik 4mo ago
  Why did it make you sad? I can see why it would make training researchers harder, but once someone attains a postdoc-level skill in mathematics research, wouldn't having a PhD-level AI assistant just boost one's ability to do more ambitious research?
  - beyonddream 4mo ago
    You should read that essay! The model is capable of producing phd level research on it's own with minimal set of prompt's. I was sad because of this paragraph: "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!
  - Koshkin 4mo ago
    > once someone attains a postdoc-level... ... they will discover that level is already crowded by LLMs
quxuejun 4mo ago
so good
anArbitraryOne 4mo ago
Good question, but I'm offended by the egregious misuse of a semicolon. My JIT compilation is throwing an error.
fatih-erikli-cg 4mo ago
Nothing. Your personal computer comes with written code. They are written by people who they live in malls. So they deny that they were using your computer and they claim that the written code is math. You pay their fastfood money when you buy a computer. Ask how a square root is calculated if you see mathematican. It is loop that starts from 0 to the answer. Computer does that.
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- balamatom 4mo ago
  Can confirm: am mole person, live at mall. Write software for fastfood which decides the fate of millions. Don't know who I work for or what qualities made me get the job.

179 points by ipnon 4mo ago

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.

generic92034 4mo ago
I wonder if AI is one means to overcome the natural limits of human knowledge aggregation [0]. 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/ https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
- morkalork 4mo ago
  Perhaps it will become like those cathedrals that took centuries and many generations of humans to build.
  - TimorousBestie 4mo ago
    Mathematics as an aggregate already is that cathedral. It is grander and more beautiful than any earthly cathedral.
  - generic92034 4mo ago
    Yes, but you (as a human) can still understand the cathedral (the building). This is not guaranteed for advanced AI work in mathematics in the future. If so, are we/they are really still adding to human knowledge, at this stage?
dataviz1000 4mo ago
LLM models can only predict the next token. 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.
- DoctorOetker 4mo ago
  Those are very strong claims, do you really believe an LLM can't be trained to solve Rubik's Cubes? 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...
  - r_lee 4mo ago
    honestly I'm pretty sure opus could solve a rubiks cube if you just gave it the layout if the sides and looped until it would solve it or even just take a picture of the thing, since they can digest visual input now
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4qshT 4mo ago
The Mathoverflow question was asked 15 years ago. The top answer says that the human community part is very important and spreading knowledge an critical thinking is valuable. 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%.
- sdenton4 4mo ago
  ...And in 1900, how many carriages were horseless?
  - ppll_12 4mo ago
    In 2026, how many people X-ray their feet at the shoe store or have watches with radium paint? Ironically, there is a shoe company pivoting to AI. My taxi driver told me buy the stock: https://www.bbc.com/news/articles/c98mrepzgj7o https://www.bbc.com/news/articles/c98mrepzgj7o
photochemsyn 4mo ago
If your motivation is being recognized as the best of the best, winning the competition, yes it’s probably a bleak world. But if you motivation is improving your own capabilities, with the metric being if you’re better know then you were last month, then it’s not a bleak world, there are many more tools available to help you learn and improve now then there were in the past.
zozbot234 4mo ago
> 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. 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.
- DroneBetter 4mo ago
  > done already in the 1990s by human-written programs that iterated through the finite casework that human thought had reduced the theorems to (four-colour theorem, FLT, etc.), which recent developments (eg. LLMs autonomously resolving Erdős problems) seem meaningfully distinct from. > human effort to make the results cleanly understandable well, perhaps loops of "derive proof through reasoning in English, formalise in Lean, use AST size of formal proof as a metric to optimise (via an LLM-guided search), translate back into English" could improve this? a lot of resources are being spent to make frontier LLMs more resistant to hallucinations via Lean, perhaps cogency will increase as a byproduct.

lokimedes 4mo ago

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.

groundzeros2015 4mo ago
Yeah I don’t think invention or technological development is inevitable or random. It’s path dependent and colored by individuals and culture.

getnormality 4mo ago

From one of the answers: > 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.

rdevilla 4mo ago
> Everything people have figured out needs to be in living form to carried on. 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.
- gosub100 4mo ago
  All LLMs do is launder other people's IP. So I don't think you invalidated any other claim.
- pegasus 4mo ago
  They are not invalidating this claim, and cannot, unless we'd actually try it out for a few generations. Which we shouldn't and won't. 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.

ucla_rob 4mo ago

Fortunately doing something novel is one of the main things llms can't do. 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.

konschubert 4mo ago
Maybe they can: https://news.ycombinator.com/item?id=48071262 https://news.ycombinator.com/item?id=48071262

elendilm 4mo ago

At the very foundation, chaining sentences together is what we call logic. 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.

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jebarker 4mo ago

“Comparison is the thief of joy.” 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.

r_lee 4mo ago
and so how exactly are you supposed to provide for your family in this scenario?
- codechicago277 4mo ago
  Could work in a patent office or something.
- jebarker 4mo ago
  By having a job. If that job is the same as your intellectual/artistic pursuit then you have to balance the needs of satisfying your employer and what keeps your motivation going over the long term. All I’m saying is that worrying too much about future achievement or “great contributions” are a recipe for burnout and disappointment.

bloaf 4mo ago

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.

nimonian 4mo ago
It's a delightful counterintuition that your gut feeling is mostly wrong: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf https://webhomes.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf 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!
- dennis_jeeves2 4mo ago
  >are usually discovered "for their own sake" Like prime numbers? (used in cryptography)
- skybrian 4mo ago
  If so that seems like an opportunity for people who want to work on applied math? There’s a big backlog of techniques that so far have not been useful.
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GuB-42 4mo ago

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.

s5300 4mo ago
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bee_rider 4mo ago
I do think this has been less pressing of a question for programmers. For the longest time there was infinite work to do, no matter your depth—implementing business logic, making frameworks more general, making sure things fit into cache lines—that was accessible to us non-geniuses. Maybe in the LLM era this sort of t-crossing and i-dotting will go away.
- zozbot234 4mo ago
  > Maybe in the LLM era this sort of t-crossing and i-dotting will go away. On the contrary, prompting LLMs creates a whole lot of newly accessible basic work.
Lerc 4mo ago
>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. 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

photochemsyn 4mo ago

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.

dunham 4mo ago

For context - the top answer was written by Bill Thurston, who was awarded a Fields Medal. (Kind of like a nobel prize for mathematics.)

vladislavp 4mo ago

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.

ivansavz 4mo ago
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). [1] https://minireference.com/static/tutorials/calculus_tutorial.pdf#page=9 https://minireference.com/static/tutorials/calculus_tutorial...

lisper 4mo ago

> 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.

LPisGood 4mo ago
Indeed, pedagogy is important to staving off the end of mathematics. 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.
- pertique 4mo ago
  There's a Scott Alexandar story that plays with this exact topic: Ars Longa, Vita Brevis [1] 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/ https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
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wcfrobert 4mo ago

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.

cryo32 4mo ago

Mathematican here. Writing software because it pays better.

lynndotpy 4mo ago

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.

baddash 4mo ago

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

siva7 4mo ago

There are many practical jobs left for mathematicians. Time to discover what you like to do with your hands.

dang 4mo ago

Related. Others? Bill Thurston's answer to “What's a mathematician to do?” (2010) - https://news.ycombinator.com/item?id=23461983 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 https://news.ycombinator.com/item?id=15578866 - Oct 2017 (25 comments) What's a Mathematician to do? - https://news.ycombinator.com/item?id=8265509 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 https://news.ycombinator.com/item?id=4419859 - Aug 2012 (1 comment) Edit: bonus relateds: https://news.ycombinator.com/item?id=43345503 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 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 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 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 https://news.ycombinator.com/item?id=12280139 - Aug 2016 (8 comments) Bill Thurston has died - https://news.ycombinator.com/item?id=4419566 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 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 https://news.ycombinator.com/item?id=982335 - Dec 2009 (5 comments)

susam 4mo ago
It's not mathematics that you need to contribute to (2010) - https://news.ycombinator.com/item?id=36744690 https://news.ycombinator.com/item?id=36744690 - July 2023 (65 comments)
- dang 4mo ago
  Inserted. Thanks!

ashley95 4mo ago

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...

_alternator_ 4mo ago
Is this not just a perspective issue? The fields we learn about in school are the ones that are in some sense mature, that have been cashed. 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.

jdw64 4mo ago

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beyonddream 4mo ago

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.

fasterik 4mo ago
Why did it make you sad? I can see why it would make training researchers harder, but once someone attains a postdoc-level skill in mathematics research, wouldn't having a PhD-level AI assistant just boost one's ability to do more ambitious research?
- beyonddream 4mo ago
  You should read that essay! The model is capable of producing phd level research on it's own with minimal set of prompt's. I was sad because of this paragraph: "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!
- Koshkin 4mo ago
  > once someone attains a postdoc-level... ... they will discover that level is already crowded by LLMs

quxuejun 4mo ago

so good

anArbitraryOne 4mo ago

Good question, but I'm offended by the egregious misuse of a semicolon. My JIT compilation is throwing an error.

fatih-erikli-cg 4mo ago

Nothing. Your personal computer comes with written code. They are written by people who they live in malls. So they deny that they were using your computer and they claim that the written code is math. You pay their fastfood money when you buy a computer. Ask how a square root is calculated if you see mathematican. It is loop that starts from 0 to the answer. Computer does that.

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balamatom 4mo ago
Can confirm: am mole person, live at mall. Write software for fastfood which decides the fate of millions. Don't know who I work for or what qualities made me get the job.