Feels like we need a new truth-state for mathematics: "Verified-but-Unproven" for results which we know to be true through formal verification, but without any coherent proof.
I would generally expect this state to resolve quickly to "Proven and Verified" (the tendency in mathematics historically has been for complicated first-proofs of results, followed by significant reductions in proof complexity as new tools and definitions become available), but may still be useful for assigning real credit to who/whatever produces reasonable proof for a brute-force-verified result.
> I would generally expect this state to resolve quickly to "Proven and Verified" (the tendency in mathematics historically has been for complicated first-proofs of results, followed by significant reductions in proof complexity as new tools and definitions become available),
A bit of an oversimplification, I think. Definitely there are some simplifications, but I wouldn't say really significant to the point where "resolve quickly" follows, along the lines of what you said.
Most complicated math is still nearly as complicated as when it was invented. Simplifications to presentation maybe, and some reduction in the proof difficulty yes, but not to the point of being significant in the way you describe. Any modern result that has been around for 20 years likely hasn't undergone a serious simplification, for example.
Last I checked (a few weeks ago, but things have been moving faster lately), the four-color theorem remains "computer proved". There has been work done on it since the original proof that has successfully simplified it, in the sense that those proofs involve less grinding through cases than the original. But it still involves grinding through more cases than can be sensibly labelled a "human comprehensible" proof, with some sort of insight into the problem space. It's still just a big set of cases being checked, and a useful final result, but no insight given in the proof.
This seems like an interesting test case to build up for AI models.
How does any of this benefit OpenAI? They already got the headlines, it's not like its going to be front page news when the mathematical community actually understands the results that have already been claimed.
I think the argument is that OpenAI should have done the work of cleaning up the proofs in the first place before dumping them onto the math community (or worked with mathematicians to guide the research process). It's as if an energy company dumped 20 tons of waste next to a village and the townspeople decided to clean it up. "How does this benefit the energy company" might not be the right question to be asking.
Uh the ultimate training data... Turning their slop into human accessible anything even if proven wrong is training data they should be paying tens of millions of dollars for.
The announcement doesn't say that the work will be using openai tools. If anything, the events over the past two months provide strong incentive to boost the capabilities of open-weight, local models.
If mathematicians didn't enjoy digging their own graves with formalization before, perhaps they will enjoy digging their own unmarked, mass graves now?
I have not been following recent developments, but does anyone know what the general sentiment among mathematicians is about these proofs: does it look like they hold up?
I know that at least two of the problems, the Unique Games Conjecture and Barnette's Conjecture, were very significant. It's probably too early to tell, but is there a feeling that these are now solved, or not?
The two most signficant number theory ones are considered solved: Quasi-Riemann and no Landau-Siegal zeros. Quanta has an article (https://www.quantamagazine.org/ai-proved-the-quasi-riemann-h...) reporting that experts agree that the Quasi-Riemann Hypothesis has been proven. The "no Landau-Siegal zeros" proof already has several manuscripts giving expositions of the proof.
The authors are all in computational mathematics and AI. This is the soft power version of the OpenAI repo to get people to work for free. Maybe call it OpenOpenAI.
How about instead, we reject the premise that OpenAI should get away with being lazy and incompetent by dumping a bunch of slop and letting researchers do all the work for free, while they still get credit?
If they want to pretend that they can do maths, they should do maths.
I personally would never participate in this. I think clarifying proofs generated by a machine is a contemptible way to spend time. A lot of people will ask, isn't the purpose of math to generate results? I don't think so. I think the purpose of math is to gain a precise and clear understanding of the world through a shared human practice of math - human only.
I know most people here will disagree because most people here are at the top of the consumption food chain of just consuming the latest and greatest of whatever the world produces, so why not have more to consume? But slowly the joy is being sucked out of the world by AI for those not at the top and being replaced the narcissistic activity of enjoying superficial sensory stimulation in a vacuum.
Leaving your “math is for humans” take alone, if the purpose of math is to “gain a precise and clear understanding of the world”, why is clarifying results a contemptible way to spend time?
But pure math is a game that only cares about the rules and definitions as you choose to define them. It's not about understanding the world at all. The application of math in science (which is about understanding the world) is usually (but admittedly not always) after the fact.
sdenton4 15 hours ago
I would generally expect this state to resolve quickly to "Proven and Verified" (the tendency in mathematics historically has been for complicated first-proofs of results, followed by significant reductions in proof complexity as new tools and definitions become available), but may still be useful for assigning real credit to who/whatever produces reasonable proof for a brute-force-verified result.
bemtevi 11 hours ago
A bit of an oversimplification, I think. Definitely there are some simplifications, but I wouldn't say really significant to the point where "resolve quickly" follows, along the lines of what you said.
Most complicated math is still nearly as complicated as when it was invented. Simplifications to presentation maybe, and some reduction in the proof difficulty yes, but not to the point of being significant in the way you describe. Any modern result that has been around for 20 years likely hasn't undergone a serious simplification, for example.
jerf 7 hours ago
This seems like an interesting test case to build up for AI models.
cadamsdotcom 14 hours ago
vishna 16 hours ago
sebzim4500 11 hours ago
reasonableklout 9 hours ago
howunfortunate 6 hours ago
The contents are actually extremely valuable, it's better to have them then not, but it still takes nontrivial expert work to sort and parse them.
simianwords 6 hours ago
thatsabadlook 8 hours ago
curt15 13 hours ago
turzmo 15 hours ago
pixl97 15 hours ago
unknownian 8 hours ago
dreijs 8 hours ago
I know that at least two of the problems, the Unique Games Conjecture and Barnette's Conjecture, were very significant. It's probably too early to tell, but is there a feeling that these are now solved, or not?
QuesnayJr 6 hours ago
athrowaway3z 15 hours ago
1293475 16 hours ago
pred_ 4 hours ago
If they want to pretend that they can do maths, they should do maths.
bemtevi 11 hours ago
I know most people here will disagree because most people here are at the top of the consumption food chain of just consuming the latest and greatest of whatever the world produces, so why not have more to consume? But slowly the joy is being sucked out of the world by AI for those not at the top and being replaced the narcissistic activity of enjoying superficial sensory stimulation in a vacuum.
A rather sad state of affairs...
_vertigo 3 hours ago
coderenegade 10 hours ago
lukax 15 hours ago
nostrebored 15 hours ago
clircle 15 hours ago
greyw 14 hours ago
reasonableklout 9 hours ago
weknowbetter 14 hours ago
greyw 13 hours ago
boothby 10 hours ago
fergonco 15 hours ago
qwr186 15 hours ago
GoWaitInTheVan 14 hours ago
aorth 4 hours ago