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.
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.
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.
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!
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.
Parents reads as a comment on the usefulness of applying mathematics to problems in the world (applied mathematics) and discovering mathematical problems that push mathematics forward (pure mathematics) in the process. Pure mathematics is incredibly important, but I’d hardly count it as useful if we need to wait centuries.
Lots of fun counter examples to this. Complex numbers were introduced in the 1600s with no practical application for almost 300 years until they were used in electromagnetism and quantum mechanics.
> 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.
In theory, sure. In practice, our society is a) not set up to value things which don’t have an immediate financial ROI, b) is valuing them less as time progresses, not more, and c) is experiencing some very serious transitions that may destroy the financial viability of devoting a lot of your time and energy to some very important things.
> 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.
Living culture is a concept that I think is quite unintuitive to modern minds. Examples of it are all around us... but it's usually blatantly missing from our "big picture" thinking.
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.
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.
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.
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 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.
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.
> honestly I'm pretty sure opus could solve a rubiks cube
honestly I'm pretty sure you made this up without any empirical evidence. I've spent several hundreds of dollars on tokens attempting to get an LLM to solve a Rubik's cube. Of course, reinforcement learning can solve a robot hand with fingers and computer vision to solve a Rubik's cube, however, that isn't the problem space.
I'm sure that a LLM could, given enough money, can be trained to solve a Rubik's cube. Currently, with all the AI companies burning a lot more money than they earn and that LLMs can't predict the consequences of their actions, there isn't any model that will solve a Rubik's cube.
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%.
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?
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?
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.
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.
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 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!
Like prime numbers? (used in cryptography)
> 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.
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.
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.
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.
It must feel similar to those who wanted to become chess or go masters after computers surpassed humanity in those games.
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...
or even just take a picture of the thing, since they can digest visual input now
honestly I'm pretty sure you made this up without any empirical evidence. I've spent several hundreds of dollars on tokens attempting to get an LLM to solve a Rubik's cube. Of course, reinforcement learning can solve a robot hand with fingers and computer vision to solve a Rubik's cube, however, that isn't the problem space.
I'm sure that a LLM could, given enough money, can be trained to solve a Rubik's cube. Currently, with all the AI companies burning a lot more money than they earn and that LLMs can't predict the consequences of their actions, there isn't any model that will solve a Rubik's cube.
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%.
Ironically, there is a shoe company pivoting to AI. My taxi driver told me buy the stock:
https://www.bbc.com/news/articles/c98mrepzgj7o
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/
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 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.
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.
https://news.ycombinator.com/item?id=48071262
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.