The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question.
The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains.
To me the potential beauty of this is that these tools might help us break through the increasing super specialization that humans in science have to go through today. Which in one hand is important on the other hand does limit the person in terms of the tooling and inspiration it has access to.
What you describe here has always been true in all sciences, but also in medicine. But both modern engineering and education runs completely counter to this. You are encouraged to stay in your niche and never look out. People with vast interested are filtered out by hiring managers.
So the crossdomain pollination that used to exist in scientists is not only not encouraged. It's also actively punished by society.
> But both modern engineering and education runs completely counter to this. You are encouraged to stay in your niche and never look out. People with vast interested are filtered out by hiring managers.
Can you explain more what you're referring to, because this has not been my experience at all. Heck, when I went to college, cross disciplinary majors were all the rage.
I think the thing that is just factually difficult is to actually become skilled in multiple different domains, precisely because the level of study/practice/rehearsal to become proficient in any individual domain keeps going up.
A long time ago you could be a Renaissance man by essentially dabbling in different fields. But today, as this article points out, you need extremely deep expertise in any one area just to understand the status quo - this proof required extremely deep expertise in two separate areas that mathematicians were surprised to be related at all.
You are making a great point here. I think it’s not just the amount of information and complexity of the domains today, it’s also human nature and emerging politics too.
Many breakthroughs come from taking an idea from one field and applying it somewhere else. But, almost every serious field is now so deep/complex/huge that humans rarely get the time, or even have enough practically useable memory, to understand and correlate multiple unrelated areas properly.
And this is where machines, such as these reasoning LLMs, can help. Because they can remember patterns across many domains and try absolutely bonker weird connections and ideas.
We, the humans still have to verify the work (at least as of now). But, the "maybe this tool, or idea, or trick, from that completely unrelated field applies here" reasoning/experimentation could become much easier.
I have always said this and will say it again: reasoning is just experimentation with a feedback loop and continuous refinement.
I’ve always been skeptical about the role of LLMs in mathematics, but this is the first time I’ve seen this argument, and I actually find it very compelling. Maybe LLMs will help us develop more horizontal understanding of the field.
It's up to us I think. We can use LLMs to generate web pages in candy crash style and end up dumper by outsourcing thinking to the machines or we can use it to expand our cognitive capabilities.
What makes me more of an optimist in this case is that people who today decide to go into these sciences are mostly people who are driven by intellectual activity so I feel they are the right ones to figure this out, probably more so than us the engineers.
The “we’s” are different. Some of us will use AI to replace human relationships and our own decision making, others of us will use it to make amazing art and invent new things.
Unfortunately, LLMs might lead to the demise of the primary institution that allows for people that are in it for the love of intellectual activity to do that activity, namely research universities. Certainly the people proposing the tech are quite opposed to the modern university.
What little intelect we have can be directed to other parts of the vast endless ocean of unknown things.
I hear some specialists (specially multi-disiplinary ones) write things they know few or no one can read. (Which is the most ironic reason for being rejected by a journal)
I recall a funny moment on irc where a truly helpful guy moaned that no one helped him when he had a (programming) question. He was very good at many programming languages and worked in some mix of high level physics and mathematics. He posted SO questions that rarely got an appologetic response from someone able to understand the code and the physics but couldnt wrap around the math. lol I hope he finally gets some help with his wizardry.
You’re making some generalizations here, but I do agree that one of the primary dangers of LLMs is destruction of institutions of higher education. If thinking power becomes cheap, who will pay the money that universities demand?
I think traditionally engineering was supposed to be the discipline that brings the breadth that science has to give up. At least that’s how I rationalized the pain I had to go through in college studying EE.
I don’t think that this model works anymore though.
Also, I love the expression “merge the horizon in”. Being a non native speaker of a language is so nice some times. Thanks!
One of the challenges I had in graduate mathematics was just trying to keep all the concepts in my brain. It doesn’t help that you end up with things like homomorphism and homeomorphism tangling one’s brain thanks to their superficial similarities. Heck, just keeping track of basic theorems and definitions is a challenge.
Yep. The thing is people (maybe because of our limited scope) just focus on the depth and not the breadth. Because this is a general purpose model - it also has PhD+ knowledge in Physics, Biology, History, etc.
I think we still don't really comprehend how much can be achieved by a single "mind" that has internalized so much knowledge from so many areas.
That's the whole point of LLM, connecting all the missing dots no single human could possible keep in working knowledge, even just for a subfield of mathematics alone. The era of polymaths is over for a reason, so we build a new one to tackle that. If LLMs can build on top of that once all remaining ones are found or if this stalls is yet to be proven, but humans stalled out there too.
There are so many research papers; just finding a solution to, say, a bio problem in a deep math paper would be a gold mine of opportunity. Very exciting times!
To me, AI feels like the morbidity of Star Trek teleportation, where it's actually copying the person at to the other end and zapping the original one out of existence. The original human never benefits from the fast transportation.
Similarly, we're creating tools to improve knowledge, but we're progressively zapping the human out of the equation. Knowledge is created for something, but it's unclear if very soon humans will be able to understand it, or really benefit from it, except billionaires, etc.
It's too bad that we're not improving humans nearly as fast as we're replacing ourselves.
Beverse they are benefitting from the financial situation of owning the ai companies that are getting pumped massive amounts of money, not from the debated usefulness of the output of the LLMs.
"leftist subtexts" such as an understanding of capitalist economics in which the people who own everything benefit from the economic activity being done underneath them.
"leftist subtexts" that treat this as a problem, and not as the single largest driver of growth in human history, to which we owe all of the material comfort we now enjoy.
Billionaires are going to benefit from AI at the expense of everyone else. That's not leftist ideology, that's just a fact. That's happened with every technology that's ever been created. It happened with the industrial revolution. Why would it be different this time?
A few years ago i had laser eye surgery and now i can see well. Something that was not possible 50 years ago. Claiming that EVERY new technology comes at the expense of normal people is an absurd statement.
Coming at your expense and benefiting you in some way are not mutually exclusive. For that matter coming at the expense of the average joe and then many years later benefiting the average joe seems to be a common theme.
What do you think of things like the changes to infant mortality or life expectancy between the industrial revolution and present day?
EG, my own oldest child needed a surgery at birth that would have been logistically impossible even 50 years ago. I'd say that she and I have benefited enormously, despite not being billionaires.
edit: I solemnly swear that the sibling comment with the strikingly similar "impossible 50 years ago" claim is a pure coincidence and that I at least am not a bot campaign. Haha.
The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains.
To me the potential beauty of this is that these tools might help us break through the increasing super specialization that humans in science have to go through today. Which in one hand is important on the other hand does limit the person in terms of the tooling and inspiration it has access to.