Tao's Rule of Thumb (which applies very well to software):
> My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.
I wonder what his views on the 4 color problem are. One can explain it as the computer checked a bunch of cases and all maps reduce to one of these cases. It doesn’t take an expert to state this.
Properly explain is an enormous grey area. Soon, I think, there will be proofs of results that are verified in Lean that are so long that no one will be able to “properly explain”. I don’t think they should be discarded.
Resolution of singularities is a famous theorem of Hironaka. Abhyankar claimed that no one truly understood the theorem. He said that he and Zariski couldn’t get through the paper with a full understanding. But everyone accepts this theorem as being correct.
We have a big problem if the only ones who understand a critical thing are machines explaining it to other machines. What happens when they start asking the LLM to write laws and judge offenses? You try to read the law but it's 30k lines of theorems. You object but robo judge says that ignorance of the law isn't an excuse. He promises you the law is sound though, his reasoning is bullet proof, and all the other machines agree. Isn't the proof enough for you? It's math and the computer checked it so that's how you know its correct.
Terence argues that explanation of results ("understanding") will be the new bottleneck in math research but I am not sure this is the real bottleneck for progress.
Understanding was critical for the field to progress when only humans were involved but if humans are not needed to make progress, I wonder if we split into two worlds: an AI math-world where amazing new results continue at a rapid pace bottlenecked only by compute/cost and a human math-world where we understand a subset of the AI math-world as a hobby (similar to Stockfish vs human chess).
Terence Tao's quote about AI's math proofs is relatable outside of pure math: "the writing very often dwells at length on trivialities while passing briefly through — or even actively obscuring — the most interesting and novel portions of the argument."
Maybe the Hitchhikers Guide to the Galaxy series was predictive in pointing out the problems of ill defined questions (The Answer to the Ultimate Question of Life, the Universe, and Everything).
Not using AI puts one at a huge disadvantage in a career setting. Ai can find deep references better than humans now, let alone actually doing the math. The challenge is knowing which problems to tackle given the cost limitations. If you have $10k to spend on tokens, you have to choose problems that can conceivably be solved within this budget.
Tao's Rule of Thumb (which applies very well to software):
> My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.
I wonder what his views on the 4 color problem are. One can explain it as the computer checked a bunch of cases and all maps reduce to one of these cases. It doesn’t take an expert to state this.
Properly explain is an enormous grey area. Soon, I think, there will be proofs of results that are verified in Lean that are so long that no one will be able to “properly explain”. I don’t think they should be discarded.
Resolution of singularities is a famous theorem of Hironaka. Abhyankar claimed that no one truly understood the theorem. He said that he and Zariski couldn’t get through the paper with a full understanding. But everyone accepts this theorem as being correct.
We have a big problem if the only ones who understand a critical thing are machines explaining it to other machines. What happens when they start asking the LLM to write laws and judge offenses? You try to read the law but it's 30k lines of theorems. You object but robo judge says that ignorance of the law isn't an excuse. He promises you the law is sound though, his reasoning is bullet proof, and all the other machines agree. Isn't the proof enough for you? It's math and the computer checked it so that's how you know its correct.
Terence argues that explanation of results ("understanding") will be the new bottleneck in math research but I am not sure this is the real bottleneck for progress.
Understanding was critical for the field to progress when only humans were involved but if humans are not needed to make progress, I wonder if we split into two worlds: an AI math-world where amazing new results continue at a rapid pace bottlenecked only by compute/cost and a human math-world where we understand a subset of the AI math-world as a hobby (similar to Stockfish vs human chess).
Terence Tao's quote about AI's math proofs is relatable outside of pure math: "the writing very often dwells at length on trivialities while passing briefly through — or even actively obscuring — the most interesting and novel portions of the argument."
If the title have said in the age of "LLMs", I might have given it a try.
Anyone else print their white papers before reading? (At least the short ones)
When I was in academia and had easy access to a good printer, I always did. I miss it now that it's easier to just read on my screen.
Maybe the Hitchhikers Guide to the Galaxy series was predictive in pointing out the problems of ill defined questions (The Answer to the Ultimate Question of Life, the Universe, and Everything).
Not using AI puts one at a huge disadvantage in a career setting. Ai can find deep references better than humans now, let alone actually doing the math. The challenge is knowing which problems to tackle given the cost limitations. If you have $10k to spend on tokens, you have to choose problems that can conceivably be solved within this budget.