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Maxwell's name being invoked here for instance implies a hundred year old foundational problem like Fermat, but it's just a recent conjecture that was inspired by reflections from the great man on his work.
* As more of the small stuff is just proven for free, the more they can be used as a basis for other proofs. If you know something is true or false for certain, that can be a significant tailwind for the much harder, much more important problems. Fermat's last theorem looks deceptively simple and invited many failed amateur attempts at solving it, but Wiles' proof drew on a diversity of seemingly-distant subfields within mathematics that were better understood.
* What are aspiring math Phd's supposed to do, now that the bar is much higher these days? The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field.
It seems plausible that the value of education will go down for the vast majority of fields and as a result less people will be getting degrees of all types.
Not a good outcome I think for humanity to be less educated, even if people are provided for when they can't get jobs... things like mathematical and scientific literacy, as well as history knowledge (which even STEM majors often receive via undergraduate degree breadth requirements), etc. I would expect strongly result in more informed and harder to deceive citizens.
We need to maintain perspective here, a PhD is essentially work done by a researcher at the least experienced, least skilled point of their career. Their primary goal is to demonstrate that they are capable of contributing to research.
They are not competing against AI to publish a counterexample to a known conjecture.
AI solving these makes me feel like mathematicians put far more importance on their work than was actually there. Many solutions seems to be tautological games, and games of logic where conjecture puzzles that few work on or care can be solved by AI which doesn’t care what it works on.
It seems to always be some form of this:
Mathematician: “Propose conjecture a and conjecture b can’t be true simultaneously”
AI: “they can”
Everyone: “ok…”
I know this might be unfair or out of ignorance but it genuinely is how this field feels today. Games of games with self importance added in.
Edit: the point I should have made is, should we be using AI to figure out what proofs MATTER now vs games of proofs?
All proofs are a form of tautology, you have to end up back at the point your theorem proposed. Math is games of logic. That's what it is.
Or it ends up never becoming useful. But you can't know that in advance
See pattern, conjecture generalization, test generalization. It's almost like empirical math. I like it and I also like mathematicians doing it the old way.
Today we can solve nontrivial open problems. What will we be able to do next year or the year after? 18months ago no one was using a coding agent seriously. Now for a large segment of the population you cannot do your job without them.
I expect to see frontier labs or startups hiring experimentalists to provide data for LLMs to analyze, pushing towards breakthroughs in areas like room-temperature superconductors and fusion.
Physics as a domain is a nightmare. Even the employment statistics are hard to understand because, like Philosophy, only the best of the best pursue it.
I've had countless friends throughout my PhD studies tell me that their decision to pursue a PhD in Physics ruined their lives. (Which is an exageration, but you get the point.)
The bottom line is that you should pusue Physics only if you still want to in the face of excessive media/reccomendations/statistics telling you not to.
Experimental physics is worrisome because as education becomes less-valued by society, there will be less funding available for research in general. Costly and elaborate 'big science' projects -- the kind that have to span multiple Presidential administrations in the US -- will be among the first to be classified as "waste," and performatively killed by legislators who earn votes by emulating the feeble-minded cultists who elected them.
I think the way forward, at least in the US, is going to involve leveraging AI to build better models to reduce our dependence on experiment. This will be true of both biology (where's the next HeLa line going to come from, once Christian nationalists complete their takeover of NIH?) and physics (ditto the next RHIC or SLAC.)
Yes, this policy amounts to eating the next generation's seed corn, but that's what Americans are voting for.
... of course, the number of crackpots is overwhelming
What is the purpose of mathematics? To be the architect of new conventions by seeing clearly past the old? If so, believing that the entire point is proving statements is a poor start. Bill Thurston was a visionary who happened to prove a great deal of what he saw, but his influence was his vision.
For those of us who like to understand every line of code we generate, and have labored for years to learn how to make best use of AI, a factor of two is a reasonable estimate for our productivity gain.
For those of us who believe mathematics is about achieving human understanding, having machines decide what's true and what isn't makes a night and day difference. Again, about a factor of two.
At the same time, most of the proofs I've looked at appear super messy and chaotic to me (while still being correct of course, so it doesn't matter). LLMs do not care about "elegance" the way human beings do, which is a big advantage. LLMs for mathematics is such a great fit on many levels. Can't wait for a significant breakthrough, prove P=NP and all hell breaks loose.
In fact, the capability of the human brain to understand complex structures and proofs is rather limited.
LLMs (hmm, I would prefer to use 'AI solver', as LLM is nowadays just a part of it) finding a complex proof can mean several things: 1) AI by its nature/construction does not have preference for simple stuff (it 'thinks' differently than human: a human will, in its search for a proof, start by exploring the 'simpler' parts of the proof space, and hence more likely find a 'simple' proof, while a AI might be more target oriented and descend deeply in depth-first-search manner to recursively solve sub-tasks, without much regard about the overall simplicity of the proof). This can be eventually solved, by subsequent 'polishing' passes, similarly as things work in human science.
2) there might simply not exist a simple/elegant proof of a given problem. The world is a complex beast. Its just our brains trying to find simple/elegant meaning/structure, even in places where there is none.
It's just a matter of time before you can post train it for elegance too. Mathematical proofs in particular can be formally verified automatically which is a big advantage.
This might not help you with finding the proof, but once you have a machine that can produce several different proofs, you can select among them and incrementally polish the best one.
I think this is the 'easier' part.
The amount of times humanity has said this and time itself was not enough of an ingredient to achieve some anticipated outcome are legion. But we filter those out and go back to making more predictions based on the current linear derivative we’re observing.
And I look forward to a single example where this happened....
How do you know they're correct if they're super messy and chaotic?
> AI "Proves" Collatz Conjecture with Lean 4 Bug
https://news.ycombinator.com/item?id=49101465
First time I heard that, and I doubt it. Don’t customers pay for output tokens? If so, why would a company specifically spend time training their LLM to generate fewer?
(Wikipedia redirects Maxwell's conjecture to Maxwell equations).
Nothing. They're still just as valid as they were before.
> For electromagnetism?
In practical terms, nothing significant. It's not going to change how anyone builds devices that use electromagnetism.
Most common names have redirects and Wikipedia is very complete. Was it just not commonly known by that name?
Having the title "The Maxwell Conjecture Is False (GPT 5.6 Sol)" instead of "The Maxwell Conjecture Is False" is editorializing
Does this work like a bug bounty program, where OpenAI pays you if you find a nice application for ChatGPT?
EDIT: Guys! Sarcasm!