5 ms·
What if the proof were incomprehensible to humans?
by hyperbovine 2y ago
What if the proof were incomprehensible to humans?
- camjw 2y agoI think that is unlikely to be the case - the classic example of a proof that human's "can't understand" is the Four Colour Theorem, but thats because the proof is a reduction to like 100000 special cases which are checked by computer. To what extent is the proof of Fermat's Last Theorem "incomprehensible to humans" because only like a dozen people on the planet could truly understand it - I don't know. The point of new proofs is really to learn new things about mathematics, and I'm sure we would learn something from a proof of Goldbach's conjecture. Finally if it's not peer reviewed then its not a real proof eh.
- hyperbovine 2y agoI think the 4-color theorem is rather different though. It reduced to a large number of cases that can in principle each be verified by a human, if they were so inclined (indeed a few intrepid mathematicians have done so over the years, at least partially.) The point of using a computer was to reduce drudgery, not to prove highly non-obvious things. Thinking back to Wiles' proof of FLT, it took the community several years of intense work just to verify/converge on the correct result. And that proof is ~130 pages. So, what if the computer produced a provably correct, 4000-page proof of the Goldbach conjecture?
- steego 2y agoIt shouldn’t count. We need to require it be able to ELI5 the proof to Goldbach’s conjecture to an actual class of graduating kindergartners.
- fanatic2pope 2y agoIf it cannot explain how it was proven, was it actually proven?
- cynicalpeace 2y agoNo. Funny how these discussions too often devolve into semantics lol.
- auggierose 2y agoFunny how people don't understand basic logic. If it is a proof in a logic, and the machine checked that proof, it is a proof, no matter that no human actually understands it. A human doesn't need to understand the proof, they just have to understand why the proof is a proof.
- moffkalast 2y agoWell... assuming a human made no mistakes setting up that logic.
- auggierose 2y agoOf course. That falls under "understanding why the proof is a proof".
- throwaway290 2y agoNow we only need to find that human that never makes mistakes and we're golden...
- deleted 2y ago[deleted]
- auggierose 2y agoLuckily, that is not necessary. You can make many mistakes, until you arrive at a logic you are happy with. Then you talk with other humans about it, and eventually you will all agree, that to the best of your knowledge, there is no mistake in the logic. If you pick first-order logic, that has already been done for you. Then you need to implement that logic in software, and again, you can and will mistakes here. You will use the first version of that software, or another logic software, to verify that your informal thoughts why your logic implementation is correct, can be formalised and checked. You will find mistakes, and fix them, and check that your correctness proof still goes through. It is very unlikely that it won't, but if it doesn't, you fix your correctness proof. If you can indeed fix it, you are done, no mistakes remain. If you cannot, something must be wrong with your implementation, so rinse and repeat. At the end of this, you have a logic, and a logic implementation, which doesn't contain any mistakes. Guaranteed.
- humansareok1 2y agoIf we can formally verify the proof then it doesn't matter. Often the implications on other problems is substantial just knowing the proof exists.