5 ms·

Good point! I'll try to think about a better way to phrase this (happy to hear suggestions)

by MCSP 2y ago

lcnPylGDnU4H9OF 2y ago
"Although mathematicians have been able to prove this, the proof is too complex to verify without computer assistance." Or some such.
valenterry 2y ago
Nah, actually I agree with you. What counts as believe and what as fact is rather abitrary. Is 2+2=4 a fact? Is global warming a fact? What about man-made global warming? Ask 100 people whether something is a fact or a believe. To top that up, it's fact that there have been "proves" that were wrong (or maybe that's just my believe? :^]) even for a long time. Hence, I think we can say that there are 4 options for a theorem: 1) Some mathematician believes the theorem is correct (but can't prove it) 2) Some mathematician believes the theorem is incorrect (but can't prove it) 3) Some mathematician believes the proof of a theorem is correct 4) Some mathematician believes the proof of a theorem is incorrect Proving that a proof is correct is kind of meaningless. At that point it's all believe anyways.
- konschubert 2y ago
  ^ Exhibit A why using "believe" is a bad choice of words. Mathematical poofs are either correct or false. There is no middle ground.
  - karmakurtisaani 2y ago
    Well.. there is. Middle ground being a very complex, but somehow convincing argument that no one can reasonably check. There was one of these cases in number theory some years ago, can't remember the details. Proofs can be only true or false, but accepting proofs is in the end a social process.
    - unclad5968 2y ago
      A convincing argument that cannot be checked is not a proof. If you want to extend the definition of proofs you're welcome to do that, but for academic mathematics the meaning of proof doesn't contain a middle ground.
      - valenterry 2y ago
        Why would it not be a proof? What is your criteria of "can be checked then"? If a proof for "sqrt(2) is not a rational number" can't be checked by a 5yo, it's still a proof no?
    - nhatcher 2y ago
      A couple come to mind * The proof of the classification of simple groups[0] * The work on topological four manifolds by M. Freedman [1] [0]: https://en.m.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.m.wikipedia.org/wiki/Classification_of_finite_sim... [1]: https://news.ycombinator.com/item?id=28471159 https://news.ycombinator.com/item?id=28471159
    - konschubert 2y ago
      > Proofs can be only true or false Yes. The fact that we don't know the truth doesn't mean there isn't one.
- blowski 2y ago
  Brilliant episode of the BBC’s In Our Time that goes down this rabbit hole. https://www.bbc.co.uk/programmes/b04v59gz https://www.bbc.co.uk/programmes/b04v59gz
ifdefdebug 2y ago
s/believe/know

Good point! I'll try to think about a better way to phrase this (happy to hear suggestions)

by MCSP 2y ago

Good point! I'll try to think about a better way to phrase this (happy to hear suggestions)

"Although mathematicians have been able to prove this, the proof is too complex to verify without computer assistance." Or some such.

valenterry 2y ago

Nah, actually I agree with you. What counts as believe and what as fact is rather abitrary. Is 2+2=4 a fact? Is global warming a fact? What about man-made global warming? Ask 100 people whether something is a fact or a believe. To top that up, it's fact that there have been "proves" that were wrong (or maybe that's just my believe? :^]) even for a long time. Hence, I think we can say that there are 4 options for a theorem: 1) Some mathematician believes the theorem is correct (but can't prove it) 2) Some mathematician believes the theorem is incorrect (but can't prove it) 3) Some mathematician believes the proof of a theorem is correct 4) Some mathematician believes the proof of a theorem is incorrect Proving that a proof is correct is kind of meaningless. At that point it's all believe anyways.

konschubert 2y ago
^ Exhibit A why using "believe" is a bad choice of words. Mathematical poofs are either correct or false. There is no middle ground.
- karmakurtisaani 2y ago
  Well.. there is. Middle ground being a very complex, but somehow convincing argument that no one can reasonably check. There was one of these cases in number theory some years ago, can't remember the details. Proofs can be only true or false, but accepting proofs is in the end a social process.
  - unclad5968 2y ago
    A convincing argument that cannot be checked is not a proof. If you want to extend the definition of proofs you're welcome to do that, but for academic mathematics the meaning of proof doesn't contain a middle ground.
    - valenterry 2y ago
      Why would it not be a proof? What is your criteria of "can be checked then"? If a proof for "sqrt(2) is not a rational number" can't be checked by a 5yo, it's still a proof no?
  - nhatcher 2y ago
    A couple come to mind * The proof of the classification of simple groups[0] * The work on topological four manifolds by M. Freedman [1] [0]: https://en.m.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.m.wikipedia.org/wiki/Classification_of_finite_sim... [1]: https://news.ycombinator.com/item?id=28471159 https://news.ycombinator.com/item?id=28471159
  - konschubert 2y ago
    > Proofs can be only true or false Yes. The fact that we don't know the truth doesn't mean there isn't one.
blowski 2y ago
Brilliant episode of the BBC’s In Our Time that goes down this rabbit hole. https://www.bbc.co.uk/programmes/b04v59gz https://www.bbc.co.uk/programmes/b04v59gz

ifdefdebug 2y ago

s/believe/know