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Mathematicians insist that in many cases the route toward solving a problem is more valuable than solving the problem itself. This may have been true, and may s

Mathematicians insist that in many cases the route toward solving a problem is more valuable than solving the problem itself. This may have been true, and may still be true for now, but this seems to implicitly assume that future generations of AI cannot be creative in similar ways that human mathematicians have been. It assumes that the absence of human comprehension during problem exploration loses a valuable aspect of mathematics that cannot be recovered by aiming future AI at solved problems in a naive environment. If we get to a future where all frontier mathematics contributions are by AI. A future where AI displays creativity in ways that expand mathematic exploration similarly to the ways humans have in the past. A future where AI explains frontier mathematics to curious humans. What will have been lost? Perhaps just "The pleasure of finding things out".

bayindirh 12d ago
> What will have been lost? Perhaps just "The pleasure of finding things out". I'll kindly disagree on this front, because when I'm walking a path toward solving a solution, I mark a lot of steps for possible diversions to other paths hence solving adjacent or different problems with the method I have at hand. Currently, AI takes us from A to B, and is improving on that front. However, the paths in science are not lines, but a trees. Methods are cross-pollinated from each other. Human intuition enables this cross-pollination. AI works with a laser focus. Human intuition and resulting wide perspective sow the seeds for solutions in many areas at once.
- Spacecosmonaut 12d ago
  I recognize that if a document were to be published tomorrow containing solutions to all formulated open problems in mathematics we will have gained very little in mathematical understanding. But surely, if we know that the dead ends of exploring a problem are valuable, we should be able to explore them even if a solution is already known. It just requires that the mathematics community reshapes itself. And it must. Two years from now people might be able to run the computation that solved NS on their Iphone. I'm sure that whatever has been discarded during the NS exploration as a dead end you would be able to rediscover using purpose built tooling in the near future. The purpose of human mathematicians in the medium term might be to explore dead ends, and to provide human insights as context to attack other problems. But whether this type of work will remain necessary in the long term im not sure.
lp4v4n 12d ago
>this seems to implicitly assume that future generations of AI cannot be creative in similar ways that human mathematicians have been It's because they can't. It's a mathematical theorem that no algorithm that "solves mathematics" can exist. >In mathematics and computer science, the Entscheidungsproblem is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an input statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936. https://en.wikipedia.org/wiki/Entscheidungsproblem https://en.wikipedia.org/wiki/Entscheidungsproblem

bayindirh 12d ago

> What will have been lost? Perhaps just "The pleasure of finding things out". I'll kindly disagree on this front, because when I'm walking a path toward solving a solution, I mark a lot of steps for possible diversions to other paths hence solving adjacent or different problems with the method I have at hand. Currently, AI takes us from A to B, and is improving on that front. However, the paths in science are not lines, but a trees. Methods are cross-pollinated from each other. Human intuition enables this cross-pollination. AI works with a laser focus. Human intuition and resulting wide perspective sow the seeds for solutions in many areas at once.

Spacecosmonaut 12d ago
I recognize that if a document were to be published tomorrow containing solutions to all formulated open problems in mathematics we will have gained very little in mathematical understanding. But surely, if we know that the dead ends of exploring a problem are valuable, we should be able to explore them even if a solution is already known. It just requires that the mathematics community reshapes itself. And it must. Two years from now people might be able to run the computation that solved NS on their Iphone. I'm sure that whatever has been discarded during the NS exploration as a dead end you would be able to rediscover using purpose built tooling in the near future. The purpose of human mathematicians in the medium term might be to explore dead ends, and to provide human insights as context to attack other problems. But whether this type of work will remain necessary in the long term im not sure.

lp4v4n 12d ago

>this seems to implicitly assume that future generations of AI cannot be creative in similar ways that human mathematicians have been It's because they can't. It's a mathematical theorem that no algorithm that "solves mathematics" can exist. >In mathematics and computer science, the Entscheidungsproblem is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an input statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936. https://en.wikipedia.org/wiki/Entscheidungsproblem https://en.wikipedia.org/wiki/Entscheidungsproblem