6 ms·

I guess we can now ask a lot of simple questions in math for whose there can be answer, but the solution itself is untenably long to be produced or understood b

I guess we can now ask a lot of simple questions in math for whose there can be answer, but the solution itself is untenably long to be produced or understood by mere mortal, not a computer or computer-like individual. Same way as we can't really reason about inner workings of machine learning model that yields useful results. Talk about transhumanism.

bhaak 11y ago
Just because his proof is long, convoluted, and cryptic doesn't mean that there won't be a simpler proof down the road. Maybe some of the tools just haven't been developed yet. Look for example at how the Greeks and Romans struggled with mathematical problems many of which are today easily solvable in high school because we have advanced techniques and a positional notation for numbers. As far as I understand it, a lot of the problem stems from the fact that Mochizuki is quite a hermit. And that's a problem if you want others to understand what you have done for years. This workshop was crucial that it showed several people that trying to understand the proof could be a worthwhile task.
- guard-of-terra 11y ago
  What if there won't be? What if, for many of questions, the only proof is as decipherable as MD5 of question?
  - bhaak 11y ago
    That's probably highly improbable. Look at how many different proofs the are for the pythagorean theorem: http://www.cut-the-knot.org/pythagoras/ http://www.cut-the-knot.org/pythagoras/ I'm not mathematician but I guess if there is really only one possible proof for a given question (Gödel might have proven that this is impossible but I don't know for sure), then the question isn't actually an interesting one. Because this would mean that the question touches only very few topics. But the abc conjecture is interesting because it has lots of links with deep questions in number theory.
    - guard-of-terra 11y ago
      Okay. What if there's a lot of proofs, but the shortest of those is still not within reach of human intellect?
      - bhaak 11y ago
        So far, we don't know if there are any proofs that are beyond the human intellect. I don't think that we currently believe such a thing to exist theoretically. Every provable conjecture should be understandable in principle by a generic human mind. Now, there for certain systems that are so complex that proofs within those systems are beyond us. Not because we couldn't understand them but because learning and applying all the rules of that system takes longer than a human mind is able to live (currently). But that's just a practical limitation, not a theoretical one.
        guard-of-terra 11y ago
        I'm talking about practical limitations. It turns out human exploration becomes impractical :(
        bhaak 11y ago
        Practical limitations can be overcome. At first it was artificial light sources, then glasses, then machines speeding up calculation and now machines that do calculations themselves. There is certainly a practical limit of what a human mind can grasp but that limit is far away and with tools, it is even much farther. We won't run out on difficult problems to solve anytime soon.
    - deleted 11y ago
      [deleted]
- randomsearch 11y ago
  > As far as I understand it, a lot of the problem stems from the fact that Mochizuki is quite a hermit. And that's a problem if you want others to understand what you have done for years. Interesting. Reinforces the idea that you should attend conferences and talk to others in your field, to ensure that you speak the same language and don't miss out on helpful advice or new tools that may (for instance) make a proof easier to follow.
erostrate 11y ago
I agree. More generally I have often wondered if pure mathematics (and theoretical physics [1]) will reach such a level of abstraction that it becomes impossible to digest all the established ideas in order to build on it. The mathematicians attending this workshop have an ability to manipulate extremely abstract concepts that is superior to the vast majority of humans, and even they can't seem to handle this proof. [1] http://www.staff.science.uu.nl/~gadda001/goodtheorist/index.html http://www.staff.science.uu.nl/~gadda001/goodtheorist/index....

bhaak 11y ago

Just because his proof is long, convoluted, and cryptic doesn't mean that there won't be a simpler proof down the road. Maybe some of the tools just haven't been developed yet. Look for example at how the Greeks and Romans struggled with mathematical problems many of which are today easily solvable in high school because we have advanced techniques and a positional notation for numbers. As far as I understand it, a lot of the problem stems from the fact that Mochizuki is quite a hermit. And that's a problem if you want others to understand what you have done for years. This workshop was crucial that it showed several people that trying to understand the proof could be a worthwhile task.

guard-of-terra 11y ago
What if there won't be? What if, for many of questions, the only proof is as decipherable as MD5 of question?
- bhaak 11y ago
  That's probably highly improbable. Look at how many different proofs the are for the pythagorean theorem: http://www.cut-the-knot.org/pythagoras/ http://www.cut-the-knot.org/pythagoras/ I'm not mathematician but I guess if there is really only one possible proof for a given question (Gödel might have proven that this is impossible but I don't know for sure), then the question isn't actually an interesting one. Because this would mean that the question touches only very few topics. But the abc conjecture is interesting because it has lots of links with deep questions in number theory.
  - guard-of-terra 11y ago
    Okay. What if there's a lot of proofs, but the shortest of those is still not within reach of human intellect?
    - bhaak 11y ago
      So far, we don't know if there are any proofs that are beyond the human intellect. I don't think that we currently believe such a thing to exist theoretically. Every provable conjecture should be understandable in principle by a generic human mind. Now, there for certain systems that are so complex that proofs within those systems are beyond us. Not because we couldn't understand them but because learning and applying all the rules of that system takes longer than a human mind is able to live (currently). But that's just a practical limitation, not a theoretical one.
      - guard-of-terra 11y ago
        I'm talking about practical limitations. It turns out human exploration becomes impractical :(
        bhaak 11y ago
        Practical limitations can be overcome. At first it was artificial light sources, then glasses, then machines speeding up calculation and now machines that do calculations themselves. There is certainly a practical limit of what a human mind can grasp but that limit is far away and with tools, it is even much farther. We won't run out on difficult problems to solve anytime soon.
  - deleted 11y ago
    [deleted]
randomsearch 11y ago
> As far as I understand it, a lot of the problem stems from the fact that Mochizuki is quite a hermit. And that's a problem if you want others to understand what you have done for years. Interesting. Reinforces the idea that you should attend conferences and talk to others in your field, to ensure that you speak the same language and don't miss out on helpful advice or new tools that may (for instance) make a proof easier to follow.

erostrate 11y ago

I agree. More generally I have often wondered if pure mathematics (and theoretical physics [1]) will reach such a level of abstraction that it becomes impossible to digest all the established ideas in order to build on it. The mathematicians attending this workshop have an ability to manipulate extremely abstract concepts that is superior to the vast majority of humans, and even they can't seem to handle this proof. [1] http://www.staff.science.uu.nl/~gadda001/goodtheorist/index.html http://www.staff.science.uu.nl/~gadda001/goodtheorist/index....