4 ms·
The confidence comes from two things, really: 1. The conjecture has been found to hold for a large number of trial cases, and 2. The conjecture has an intuiti
by mrbrowning 13y ago
The confidence comes from two things, really:
1. The conjecture has been found to hold for a large number of trial cases, and
2. The conjecture has an intuitive aura of rightness.
A good example is the 3n + 1 conjecture:
http://en.wikipedia.org/wiki/Collatz_conjecture http://en.wikipedia.org/wiki/Collatz_conjecture
Anyway, obviously point two is extremely subjective, but I think it's the crux of why long-standing conjectures are interesting. Why should something feel correct without necessarily being so? If it does turn out to be correct, what about its nature put people on its scent trail? Mathematics so often seems to be a kind of hermetic universe unto itself, so part of the beauty of conjectures is that they're kind of a contact point between that universe and human congnition.
- Someone 13y agoWhere's the intuitive arena of rightness for the Collatz conjecture? I would put an 'or' between #1 and #2 (with #1 being less preferred; why would anybody start checking whether P holds for a large number of trial cases without having a hint of #2? (How did Collatz came up with this conjecture?)), and add a #3: the conjecture isn't easily disproven ('easily' being subjective, but the other two are subjective, too)
- danbruc 13y agoWhile working on some problem or just playing around (with numbers) you could discover something by accident. Noticing the Collatz conjecture for only two numbers is probably enough to cause some curiosity and then you try some more examples and not much later you have a conjecture without any idea why it could be true.