We must be using different meanings of the word “evidence”, because it seems under your definition consequentialist mathematicians would be completely apathetic.
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.
To the extent that the research is worth pursuing, yes; but no mathematician treats such a conjecture in the same way as a proven result.
We must be using different meanings of the word “evidence”, because it seems under your definition consequentialist mathematicians would be completely apathetic.
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.