181–190 of 265 posts
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This is probably part of why machines are doing so well at counterexamples. They have no aesthetic commitment to the conjecture and no embarrassment about producing something ugly
This reminds me of the Go Grandmaster speaking out after losing to AlphaGo, that the model has no sense of "aesthetic play", as long as it would lead to a win within the rules.
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Inspiring? Because of the twin prime conjecture success following his time in the wilderness? I suppose so. I'm tired of tales like this in academics though. That's not a criticism of you for telling the tale, I'm just so tired of this kind of thing in academics in general. So, so, so much politics and public reputation management. Zhang should have never had to suffer like that. As my own research has drifted more i…
Planck's Principle: 'Science advances one funeral at a time.' [1] In his exact words: "A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it... An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happen…
If you are not yet old, remember this - it is very likely you will old in a few years.
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proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions. and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
> and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians Considering ChatGPT was released only three and half years ago, and LLMs could do high school math only less than two years ago, I think this "for now" will not last very long.
Or are they approaching an asymptotic limit?
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This reminds me of the Go Grandmaster speaking out after losing to AlphaGo, that the model has no sense of "aesthetic play", as long as it would lead to a win within the rules.
I thought there was a lot of buzz about AI creativity after the infamous move 37 in that series?
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Proof fix-ups are quite common, but if it was not possible, then no, not for that topic.
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". Probably there are understandable reasons for that... But I think "negative science" is really important, and soft results like "we tried that for a long time and it…
Typically, you can describe that in the form of a partial result (see for example https://en.wikipedia.org/wiki/Goldbach%27s_conjecture#Partia...) or as another conjecture.
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This is probably part of why machines are doing so well at counterexamples. They have no aesthetic commitment to the conjecture and no embarrassment about producing something ugly
This reminds me of the Go Grandmaster speaking out after losing to AlphaGo, that the model has no sense of "aesthetic play", as long as it would lead to a win within the rules.
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Proof fix-ups are quite common, but if it was not possible, then no, not for that topic.
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". Probably there are understandable reasons for that... But I think "negative science" is really important, and soft results like "we tried that for a long time and it…
There is a related tradition of making conjectures about things that you can not prove, and being known for having made such conjectures. Conjectures along with definitions and problem statements are incredibly important in mathematics. But usually they are introduced in the context of some other publishable work.
Long story short this is much less impressive than it was sold to be. Basically they had mathematicians combing through long winding chains of thought (incidentally: you wouldn't have access to that reasoning) and cleaning it up and making it coherent. That doesn't mean its unimportant, but we're being gaslit about the amount of human steering and human effort that went into this.
A plea: please stop upvoting hype that comes from these labs. It takes time to evaluate their claims and they're always less impressive than claimed.
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The term she mistook for meaning a death/execution had been recorded was "death recorded". So it's a mistake that could be made similarly to how one might confidently assume that a "public school" is a school which is a school that is either government funded or open to anybody in the local area. More generally I think people are inappropriately confident in general. When you go through history just about every centu…
You have to be confident by default or you'll never get anything done. I always find myself stuck in analysis paralysis and then I look back at something like the history of Minecraft and realise he wrote the whole game (minus content additions) from scratch in only about two years, and all the updates I remember not liking were just another Tuesday for him as there had been similar updates every few weeks during tha…
Crippling doubt can happen. Blind confidence has its own hazards. You only need to be confident that you can proceed, in spite of having provisional, incomplete knowledge. "But if you grip to your beliefs too tightly, you will not recognise The Truth when it comes knocking on your door." (forget who said that)
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Those are all good questions, but I don't really understand what alternative you or the OP are looking for actually resolving an untrue conjecture, besides a counter example.
There are many cases where it's possible to prove that counterexamples must exist, without identifying a specific example. This kind of proof provides more insight into the problem than simply finding a counterexample.