I’m sorry, that made no sense to me. How about I try to restate what I was trying to say?
Mathematicians believe in mathematical results on a tiered scale: first come all the proven results, and next come all the unproven results they believe are true. Testing the first 10^6 cases of a conjecture meant to apply to all integers puts it pretty high up there in the second tier, but not as high as the first tier.
Arguably, for a result to become a theorem it has to become widely accepted by the mathematical community, at which point the proof is almost certainly correct. If I read a paper that proves a result nobody really cares about, I’m more careful about believing it.
Of course, you can assume a conjecture and use it to prove a theorem; then the theorem you proved is a valid result of the form “if this conjecture is true, then”. However, a stronger result that doesn’t assume the conjecture is better.
So when you said what I quoted in the great-grandparent, that is, “[10^6 cases aren’t] going to be accepted as … particularly valid evidence”, you meant that it would be accepted as evidence, that is, it belongs to your second tier of belief? That’s what’s bothering me.
Sorry, I guess at that point I was just thinking of the first tier of belief when making that comment.
But I think it’s also true that most notable conjectures have, in addition to verification of some cases, some sort of handwavy reason why we would believe them, such as a sketch of a proof but with many holes in it, and this is at least important as the other kind of evidence.
I’m sorry, that made no sense to me. How about I try to restate what I was trying to say?
Mathematicians believe in mathematical results on a tiered scale: first come all the proven results, and next come all the unproven results they believe are true. Testing the first 10^6 cases of a conjecture meant to apply to all integers puts it pretty high up there in the second tier, but not as high as the first tier.
Arguably, for a result to become a theorem it has to become widely accepted by the mathematical community, at which point the proof is almost certainly correct. If I read a paper that proves a result nobody really cares about, I’m more careful about believing it.
Of course, you can assume a conjecture and use it to prove a theorem; then the theorem you proved is a valid result of the form “if this conjecture is true, then”. However, a stronger result that doesn’t assume the conjecture is better.
So when you said what I quoted in the great-grandparent, that is, “[10^6 cases aren’t] going to be accepted as … particularly valid evidence”, you meant that it would be accepted as evidence, that is, it belongs to your second tier of belief? That’s what’s bothering me.
Sorry, I guess at that point I was just thinking of the first tier of belief when making that comment.
But I think it’s also true that most notable conjectures have, in addition to verification of some cases, some sort of handwavy reason why we would believe them, such as a sketch of a proof but with many holes in it, and this is at least important as the other kind of evidence.