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sniffnoyHere's something that I wonder if it seems counterintuitive to non-mathematicians.
Every now and then you see something of the form, well, we can't prove conjecture X, nor can we prove conjecture Y, but we can prove that one of the two must be true[0]. (Here and throughout I'm supposing that conjectures X and Y are both generally believed to be true.) And this is always just really disappointing, because what are you going to do with that? Simply knowing that at least one of the two is true is really unsatisfying -- whichever one might be true, it gives you no idea why.
(Note -- I'm speaking here of statements of the form "Either (complete statement 1) or (complete statement 2)", not statements of the form "For every x, either x has (property 1) or (property 2)", which are a different sort of beast and often quite useful, if you can indeed show such a thing. Kind of like how the material conditional doesn't seem weird at all when it's "For all x, P(x) implies Q(x)" instead of just "P implies Q". Well, unless you never got comfortable with the empty set, then I guess it seems weird to you either way. In which case, you should do that, because seriously, it's just a set with no elements.)
Anyway. The part I'm wondering if it seems counterintuitive to some people, is that a result of the form "Either conjecture X is true, or conjecture Y is false" is much less disappointing, because it's saying "If Y, then X"; it's a conditional proof. I mean, this is obvious, I just wonder if some people might find it strange because it's better despite appearing from the form of it to be less in the direction of things believed to be true.
[Unrelatedly: What else is in the file? Well there's any number of things I've meant to write, but I only started keeping this recently, so here's a few things I've actually bothered to write down in there:
[Unrelatedly: Unprotected an old entry due to time.]
-Harry
[0]Constructivists throw tomatoes or something at this point.
Every now and then you see something of the form, well, we can't prove conjecture X, nor can we prove conjecture Y, but we can prove that one of the two must be true[0]. (Here and throughout I'm supposing that conjectures X and Y are both generally believed to be true.) And this is always just really disappointing, because what are you going to do with that? Simply knowing that at least one of the two is true is really unsatisfying -- whichever one might be true, it gives you no idea why.
(Note -- I'm speaking here of statements of the form "Either (complete statement 1) or (complete statement 2)", not statements of the form "For every x, either x has (property 1) or (property 2)", which are a different sort of beast and often quite useful, if you can indeed show such a thing. Kind of like how the material conditional doesn't seem weird at all when it's "For all x, P(x) implies Q(x)" instead of just "P implies Q". Well, unless you never got comfortable with the empty set, then I guess it seems weird to you either way. In which case, you should do that, because seriously, it's just a set with no elements.)
Anyway. The part I'm wondering if it seems counterintuitive to some people, is that a result of the form "Either conjecture X is true, or conjecture Y is false" is much less disappointing, because it's saying "If Y, then X"; it's a conditional proof. I mean, this is obvious, I just wonder if some people might find it strange because it's better despite appearing from the form of it to be less in the direction of things believed to be true.
[Unrelatedly: What else is in the file? Well there's any number of things I've meant to write, but I only started keeping this recently, so here's a few things I've actually bothered to write down in there:
- Things Eliezer Yudkowsky says that really do seem nutty to me
- The landscape of nonstandard mathematics is weirder than I realized
- A few quick notes on video game music
[Unrelatedly: Unprotected an old entry due to time.]
-Harry
[0]Constructivists throw tomatoes or something at this point.
no subject
Date: 2011-09-25 12:15 am (UTC)Also, I'm interested in hearing about 1 and 3 from the file.
no subject
Date: 2011-09-25 02:46 am (UTC)Perhaps an example would help. I think the actual disjunction that inspired this post was this one from Dick Lipton's blog. Go look. Because what can you do with the statement that *at least one* of those is true? As he points out, it can be thought of as "if one is false, the other is true"; but the problem is that probably they're both true.
Note here that I'm only talking about disjunctions over sentences in the logical sense, i.e., complete statements, not properties. IMO this sort of nonconstructive stuff seems markedly less weird when applied to properties quantified with a "for all". E.g. I don't find the Brouwer fixpoint theorem, the classic example of nonconstructiveness, unsatisfying.