This was a video I saw that covered this idea well: https://www.youtube.com/watch?v=KqiDqJMxg6o
I find this argument difficult to understand. As a math student in college with no ambitions (or realistic opportunity or aptitude) to pursue original contributions to the field, everything I learned was by definition already solved. Nevertheless I was able to - required to, in fact - work through each solution in order to develop understanding. And I did in fact learn by this method, as countless other students did before and after me.
So when I hear this argument from mathematicians, all I can hear is that they are frustrated at being out back into the position of being students, where the end result of the hard work of understanding is just the understanding itself without the prize of a new discovery. If they want to find the solution without reading it, they are still very much able to do so, just as their students do every day.
Difficult problem -> Developing new branches of mathematics to solve said problem -> solving problem, or failing to do so, it doesn't matter -> aforementioned new branches of mathematics are used to solve practical, real-world problems that benefit industry and economy.
The point of solving the problem is not the prize. The point of solving the problem is not even solving the problem. It is developing new methods and deepening understanding to drive useful applications.
Knowing that a solution exists is not an impediment to deepening your understanding of a problem.
For example: Knowing that there is a proof of Pythagoras’ theorem is not an impediment to thousands (millions?) of geometry students developing the intuition that comes from drawing the famous diagram where the number of squares visually add up to one another.
What about the existence of an AI-generated proof prevents a mathematician from working through the problem on their own to deepen their understanding, as if the proof didn’t exist in the first place?
That’s what is hard to understand.
I'm baffled that you find the mechanics of human motivation hard to understand, (presumably ;) being a human yourself.
Studies have shown that people that know a certain piece of knowledge is stored somewhere, are less likely to remember said knowledge.
The sense (thrill?) of exploration and discovery can in fact be part of a person's motivation. The popularity of open-world games (often derided as 'walking simulators') are a testament to that. Do you really not see how that could be undermined by knowing that a problem that interests you is already solved?
Basically all natural sciences work that way. Knowing the truth of an apple falling down a table when rolled over the edge is the beginning of discovery, not the end of it.
Mathematical discovery also worked more like theoretical physics 200 years ago. The modern way of formalization in maths might be dead now in some ways, but it's also only 100 years old. There's still many ways to do modern maths and have a lot of fun with it that are not taken away by AI.
it is qualitatively different
> If a set E in R^k has one of the following three properties, then it has the other two:
(a) E is closed and bounded.
(b) E is compact.
(c) Every infinite subset of E has a limit point in E.
Likewise, when given an exercise of the form "Prove P" or even "Prove or disprove P," you can be pretty certain 1. it is doable and 2. it is doable using techniques/theorems discussed in the course/book you're working through. But people are perfectly happy to do this for several years.
But that’s exactly it! The mathematicians are framing this as being bad for human understanding of math, but it’s obviously really about their own motivations for developing that understanding! Which, again, goes back to the obvious conclusion that what they have lost is the extrinsic motivation for being first, not the intrinsic motivation of developing understanding.
That is totally fine! I wouldn’t be motivated to do my job if I wasn’t extrinsically rewarded for doing so. But I am also not feeding anyone a line of BS about how it’s actually some other higher-minded motivation driving my work.
I’m empathetic to the instability of the extrinsic rewards for doing math research as a profession in the current moment. I am not at all empathetic to an attempt to manipulate me into thinking that the real problem is something other than that.
It’s like an entrepreneur complaining that they can’t build a business that would better humanity because of high taxes. That’s fine - but the motivation lacking there is obviously making money, not bettering humanity.
When you're learning/internalizing existing knowledge, you tread a defined path skipping unnecessary stops and diversion and heading instead directly to the destination. The destination, not the journey, is the point.
When you're discovering new knowledge in the form of proving/disproving, there's no defined path to follow, so there can be quite a bit of backtracking and diversions and detours into other areas you weren't planning on going initially and while there might be some well-defined paths to follow, you might have to cut your own path through uncharted desert or jungle.
All those diversions and detours make the journey just as if not more important than the destination because they create new destinations of their own for future travel that would not have been spotted or noticed for a while longer (if ever) otherwise.
So to bring it back to math, transitioning from having to develop proofs for theorems to validating/learning extant proofs completely changes the scope of future discovery. Everyone will tread the exact same path to a proof/theorem and then stand in the same place to look for future destinations and then use AI to build them a path to that new destination and then everyone will follow the same path to the new destination and repeat the process, so the chances of finding something curious along the way plummets. And something curious could be an entirely new field of math, or maybe just a new problem that isn't relevant to the intended destination but gives a view of many new never before seen destinations for future trips.
Demis Hassabis has talked about this a bit in regards to theoretical physics and it's an idea that I think I agree on. Every once in awhile we randomly get a super genius who's brain is seemingly able to conceptualise or visualise beyond the state of the art. But at some point I think we have to accept that we are just animals and inherently limited by our brains (and means of observing the physical world).
The scary part isn't that AI is solving the unsolved problems that will force people to act like students again. The scary part is that we may not be able to understand the work needed to get to the solution at all.
Obviously we're not quite to that level yet but it's entirely possible we do get there. And on the optimistic side ~ maybe that's what we need: better tools.
I see no reason whatsoever to expect that humans have any kind of limit to what we can understand as a species. We've had more unthinkable and incomprehensible discoveries than anyone cares to count. They become understood, normalized, and folded into our scientific canon. We went from inventing airplanes to landing on the moon in a matter of decades. We went from banging rocks together to quantum physics and string theory. I think we'll be alright.
But there's no guarantee that such "low IQ" intermediate steps exist. We do well with many classes of problems because of divide-and-conquer. But not all problems can be solved through divide-and-conquer.
For a rough analogy, see how AI plays chess and intermediate board positions often don't make sense to us. Imagine now an enormous chess-like game played in an enormous chess board where a match lasts longer than a lifetime and none of the intermediate steps make sense to any human.
> But closer examination reveals that OpenAI’s proof builds its unnatural force in a totally distinct way. The proof’s similarities to Buckmaster and Alpöge’s progress are superficial...
https://www.scientificamerican.com/article/openais-blockbust...
So I think more time needs to pass before a call can be made.
An interesting thought experiment is this: if we were to go back in time 150 years and tell our ancestors that Newton is just an approximation and give them a 5,000 page book about Special and General Relativity, QM and QFT, would we be robbing them? Initially the whole thing would be incomprehensible, but eventually they would chew through it and make sense of it. I don't really know the answer (and perhaps the analogy is not great since we're quite attached to our physics heroes from the last 150 years).
there is a difference between "knowing" and "knowing about". "knowing" is having it deeply ingrained into one's intuition by working at a solution over years, often without progress in the immediate sense, and often without success in the absolute sense either.
Maybe post-LLMs mathematicians will work on problems/topics that even ChatGPT can't solve? Like, P=NP remains unsolved, we can work on that. Sure, it's possible that it will get solved tomorrow by an AI or somebody else, but that possibility was always there.. Or, to be more practical, one of the claimed OpenAI results is that integer multiplication is sub n×log(n), so assuming it stands, now there's a new question: "What is the actual lower bound for integer multiplication?"
I have no proof, but I believe (i) LLM progress will converge at some point, (ii) that point will not be so far off from human intelligence, since they're trained on human intelligence, so.. not all problems will be solved/solvable by LLMs. But they'll be fast(er), like calculators are faster than us.
This is just what is happening. Techniques discovered by AI have already been used to solve other problems, or to extend the AI-discovered solutions.
I remember a recent "100 hundred mathematicians on AI" post recently. Some of them appeared truly excited even if the majority wasn't. Who's right? Who's biased? I don't know.