I wonder if we’ll end up in a weird situation a decade from now, where every conjecture will collapse into one of two states: counterexample found by AI, or assumed to be true.
> No way. Mathematicians don't assume things to be true even if there are no counterexamples.
It depends: there do exist some conjectures for which there don't exist any proofs, but a huge amount of results assuming this conjecture (which is very near to assuming that the conjecture is true). Examples are:
- Riemann hypothesis [1]
- Generalized Riemann hypothesis (GRH) [2], and potentially also Extended Riemann hypothesis (ERH) [3]
bee_rider · · focus · HN ↗
vouaobrasil · · focus · HN ↗
scrubs · · focus · HN ↗
bmitc · · focus · HN ↗
It can be productive to assume something is true and see what happens.
aleph_minus_one · · focus · HN ↗
It depends: there do exist some conjectures for which there don't exist any proofs, but a huge amount of results assuming this conjecture (which is very near to assuming that the conjecture is true). Examples are:
- Riemann hypothesis [1]
- Generalized Riemann hypothesis (GRH) [2], and potentially also Extended Riemann hypothesis (ERH) [3]
- Standard conjectures on algebraic cycles [4]
- perhaps Schanuel’s conjecture [5]
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[1] <a href="https://en.wikipedia.org/w/index.php?title=Riemann_hypothesis&oldid=1375292396" rel="nofollow">https://en.wikipedia.org/w/index.php?title=Riemann_hypothesi...
[2] <a href="https://en.wikipedia.org/w/index.php?title=Generalized_Riemann_hypothesis&oldid=1370923804#Generalized_Riemann_hypothesis_(GRH)" rel="nofollow">https://en.wikipedia.org/w/index.php?title=Generalized_Riema...
[3] <a href="https://en.wikipedia.org/w/index.php?title=Generalized_Riemann_hypothesis&oldid=1370923804#Extended_Riemann_hypothesis_(ERH)" rel="nofollow">https://en.wikipedia.org/w/index.php?title=Generalized_Riema...
[4] <a href="https://en.wikipedia.org/w/index.php?title=Standard_conjectures_on_algebraic_cycles&oldid=1374346010" rel="nofollow">https://en.wikipedia.org/w/index.php?title=Standard_conjectu...
[5] <a href="https://en.wikipedia.org/w/index.php?title=Schanuel%27s_conjecture&oldid=1375037872" rel="nofollow">https://en.wikipedia.org/w/index.php?title=Schanuel%27s_conj...