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Here is an old column of his, "New Architecture Needed"

https://www.pcmag.com/archive/new-architecture-needed-32570

I wonder whether when the fact that AIs have started solving conjectures will enter the training data, they will become more confident in their abilities.

Because I think the median reader would expect this sort of thing to come up at some point in the story.

I guess there's no tradition for publishing "negative results" in mathematics? By that I mean not proof of something negative, but rather, "we tried this thing for ages and couldn't get it to work, but we couldn't prove that it could never work either".

Indeed there isn't such tradition. I have one or two results like that -- proofs that some proof strategies cannot work because some object does not exist, but since that object would not be interesting for anyone not trying that particular proof strategy for that particular (already obscure) problem, one cannot publish.

I personally know a story in this vein with a (sort of) happy ending.

A PhD student discovered that a result of his professor would imply the solution to a big conjecture in another field. The people in that field then analyzed the prof's result and found that the proof was flawed. The student was still allowed to graduate based on this since the finding of the connection between fields was brilliant. Then he quit academia (not because of this story, he had planned it before). Then a year later the prof figured out how to fix the flaw in his proof and published a paper with his former student, thus solving the conjecture. The two are still on good terms, writing papers together.

Thanks! It makes sense.

It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample.

Hm, as a mathematician, my experience feels opposite. A proof would be an adaptation of a proof I know, some tweaking it here and there. A counterexample would require some deep understanding of the structure of the objects involved, which frequently is beyond my comprehension.

But probably this is because I think of quite abstract objects which are harder to grasp. For numbers or polynomials, this would be the other way round.

You mean proof by contradiction, which is something different.

Exactly, yes!

And the conjecture was for a specific class of processes.

And Fable found an example of one concrete* process in that class and three concrete inputs (two were enough of course) giving the same output.

*Concrete here means given by a finite string of characters

January 2025, release of DeepSeek R1, the first open reasoning model. There was a lot of panic then that it was done with very few resources.

Yeah, in my university there were two AI professors, one did neural networks, and one did search stuff, and it depended on whom you are assigned to.

In 2011, I took an AI course at my university and it was all perceptrons and neural networks.

Agriculture is pretty much the antonym of industry.

Space exploration.

To the author: for the absolute Galois of Q_p problem, the link is wrong.

Which is a metaphor for life.

I also had this sort of thoughts when finishing my master's degree. I guess what breaks the cycle is that proofs (like other artefacts in other human activities) deliver aesthetic bliss.

And Kim Jong the Second, which was confusing since he was actually the second Kim.

This is sad, almost as sad as the Deathly Hallows pre-release leak.

Wikipedia says Hong Wang while acknowledging that the native form is Wang Hong and that they are using the Western name order.

The title is misleading because it is suggests it is about a colony of the US (like Phillipines), not a colony on territory which is now US.

See, you are already confused: it's Sol/Terra/Luna, descending order by diameter. One is tempted to put the Sun and the Moon before the Earth, as those are "celestial".

Indeed, after reading the book I was surprised that the movie has explicit frontal nudity.

I think the main problem is whether intelligence is a computable function (or at least approximable by ones, like AIXI is), and then whether it's of the form that NNs implement (linear algebras plus sigmoids and all that jazz).

Yeah, I am aware of this statelessness.

This is what I was referring to:

https://www.cnbc.com/2025/08/19/sam-altman-on-gpt-6-people-w...

It's marketing speak, but the goal is clearly there, no idea how achievable.

Indeed. Any meaningful AGI/ASI will have to have a form of memory / continual learning. Sam Altman said last year that this will be the focus for GPT-6.

The whole "soul.md" stuff today is a poor approximation to that. But I wonder whether it will grow into it, like chain of thought prompting grew into reasoning models.

These stories are common in math, e.g. these recently happened to me, a lowly mathematician:

1) Two and a half years with no reply from a journal (not even to emails I sent that I'd like to retract the paper so I could send it somewhere else). Then suddenly they tell me the paper is accepted.

2) One year with no reply. Then, my "anxious" collaborator sends them countless emails and gets redirected from person to person and finally an editor tells us that they decided almost immediately to reject our paper but they didn't tell us because "they hate giving bad news".

These were not top journals like Annals, but decent, prestigious ones, from whom you'd expect some professionalism.

Thank you for spelling this in detail!

One thing I might add is that not all programs can be proved to be correct for the simple reason that not all purposes of a program can be mathematically specified. For example, for (even "closed world" domain programs like) a chess engine, the one thing that matters (in the absence of a complete solution of a game like there exists in checkers) is "can beat world champions", which can only be tested empirically. Or sometimes, e.g. business logic software, the purpose can be mathematically specified but not in a simpler way than the code itself.

I am a mathematician, and I was never the kind to like to struggle by working on problems, but I developed a lot of intuition by 1) thinking deeply about definitions and proofs and why are they this way and not another 2) reading a lot of blogs and expository papers by great mathematicians, even (the more philosophically minded) mathoverflow q&a's (so I absorbed their way of thinking unconsciously). For example, I tell my students to read all 300 of John Baez's This Week's Finds posts [1] and they will deeply understand more math than 99% of their peers.

This is not the "standard" advice that usually gets peddled but for me it "worked".

[1] https://math.ucr.edu/home/baez/twf.html

Maybe GPT-6 will not write poorly unless asked to.

The content is good, but the LLM feel is jarring.