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I recommend Terrence Tao's commentary on such a proof : https://mathstodon.xyz/@tao/117219101339291693 https://mathstodon.xyz/@tao/117219101339291693 Key quote

I recommend Terrence Tao's commentary on such a proof : https://mathstodon.xyz/@tao/117219101339291693 https://mathstodon.xyz/@tao/117219101339291693 Key quote : "Solving the problem by purely AI-powered methods [would be a] net negative for the progress of mathematics."

porridgeraisin 26d ago
The for-case for this type of method is that this is economies of scale for mathematics. We are basically mass manufacturing math. Just like you have just 100 designers for a product selling millions of units, you will now need 100 mathematicians to make millions of advancement. Yes you have factory workers, but if we are being realistic they have negative leverage in the world and the analogue of that is not something most of today's mathematicians would want to do. They would want to be in the 100. Like Tao says, each advancement is now significantly less useful since it yields fewer usable objects. However, we will get many many advancements. Is the tower made with many worse bricks better or worse than the tower made with a few amazing bricks? Depends on the tower. And time will tell. For some fields of math and some of it's usecases, economies of scale will be positive ROI overall. In others it won't. But we will know which is which only after it's been fully scaled up, which will take 10-15y in my estimate. Some feel that in the majority of usecases it is negative ROI, some feel the other way, but that opinion is for practicing mathematicians like Tao to hold. Also, some opinions on either side are held in the context of a particular field or practice, and should not be interpreted generally.
- treyjshaffer 26d ago
  This isn't a useful analogy. His point is that the millenium problems should be treated as interesting goals where the journey is the purpose and where the end doesn't matter so much. We don't care about having an incomprehensible solution to the NS so much as having an elegant solution after many of subfields of math are built up in order to obtain that elegant solution.
  - porridgeraisin 26d ago
    I agree on that, my post was not meant to be opposing this.

porridgeraisin 26d ago

The for-case for this type of method is that this is economies of scale for mathematics. We are basically mass manufacturing math. Just like you have just 100 designers for a product selling millions of units, you will now need 100 mathematicians to make millions of advancement. Yes you have factory workers, but if we are being realistic they have negative leverage in the world and the analogue of that is not something most of today's mathematicians would want to do. They would want to be in the 100. Like Tao says, each advancement is now significantly less useful since it yields fewer usable objects. However, we will get many many advancements. Is the tower made with many worse bricks better or worse than the tower made with a few amazing bricks? Depends on the tower. And time will tell. For some fields of math and some of it's usecases, economies of scale will be positive ROI overall. In others it won't. But we will know which is which only after it's been fully scaled up, which will take 10-15y in my estimate. Some feel that in the majority of usecases it is negative ROI, some feel the other way, but that opinion is for practicing mathematicians like Tao to hold. Also, some opinions on either side are held in the context of a particular field or practice, and should not be interpreted generally.

treyjshaffer 26d ago
This isn't a useful analogy. His point is that the millenium problems should be treated as interesting goals where the journey is the purpose and where the end doesn't matter so much. We don't care about having an incomprehensible solution to the NS so much as having an elegant solution after many of subfields of math are built up in order to obtain that elegant solution.
- porridgeraisin 26d ago
  I agree on that, my post was not meant to be opposing this.