Would you mind explain in a little more detail why you say a person who has seen Kempe’s flawed proof should have higher confidence than one who has not? Do you mean that it’s so emotionally compelling that one’s mind is convinced even if the math doesn’t add up? Or that the required (previously-hidden) premise that allows Kempe to ignore the degree 5 vertex has some possibility of truth, so that the conclusion has an increased likelihood of truth?
Explain better why you say a person who has seen Kempe’s flawed proof should have higher confidence than one who has not.
Hmm, I’m not sure how to do so without just going through the whole proof. Essentially, Kempe’s proof showed that a smallest counterexample graph couldn’t have certain properties. One part of the proof was showing that the graph could not contain a vertex of degree 5. But this part was flawed. But Kempe did show that it couldn’t contain a vertex of degree 4, and moreover, it showed that any minimal counterexample must have a vertex of degree 5. This makes us more confident in the original claim since a minimal counterexample has to have a very restricted looking form.
Replying to the fixed end here so as to minimize confusion:
What you are saying regarding multiple distinct proofs of a claim is true according to some informal logic, but not in any strict mathematical sense. Mathematically, you’ve either proven something or you haven’t. Mathematicians may still be convinced by scientific, theologic, literary, financial, etc. arguments of course.
Well, yes but the claim I was addressing was that the claim you made that “encountering a single incorrect premise or step means that the conclusion has zero utility to the Bayesian” which is wrong. I agree that a flawed proof is not a proof.
And yes, the logic is in any case informal. See my earlier parenthetical remark. I actually consider the problem of confidence in mathematical reasoning to be one of the great difficult open problems within Bayesianism. One reason I don’t (generally) self-identify as a Bayesian is due to an apparent lack of this theory. (This itself deserves a disclaimer that I’m by no means at all an expert in this field and so there may be work in this direction but if so I haven’t seen any that is at all satisfactory.)
“encountering a single incorrect premise or step means that the conclusion has zero utility to the Bayesian” which is wrong
I think you are assuming I count a dubious premise as an incorrect premise.
Obviously, a merely dubious premise allows the conclusion to have some utility to the Bayesian.
I think you are assuming I count a dubious premise as an incorrect premise. Obviously, a merely dubious premise allows the conclusion to have some utility to the Bayesian.
Really? Even incorrect premises can be useful. For example, one plausibility argument for the Riemann hypothesis rests on assuming that the Mobius function behaves like a random variable. But that’s a false statement. Nevertheless, it acts close enough to being a random variable that many find this argument to be evidence for RH. And there’s been very good work trying to take this false statement and make true versions of it.
Similarly, if one believes what you have said then one would have to conclude that if one lived in the 1700s that all of calculus would have been useless because it rests on the notion of infinitesimals which didn’t exist. The premise was incorrect, but the results were sound.
Incidentally, as more evidence, apparently this AC0 conjecture has just been proved true by Ben Green (rather, he noticed that other people had already done stuff that had this as a consequence, which the people asking the question hadn’t known about).
Ok, I need to refine my description of math a bit. I’d claimed that an incorrect premise gives useless conclusions; actually as you point out if we have a close-to-correct premise instead, we can have useful conclusions. The word “instead” is important there, because otherwise we can then add in a correct contradictory premise, generating new and false conclusions. In some sense this is necessary to all math, most evidently geometry: we don’t actually have any triangles in the world, but we use near-triangles all the time, pretending they’re triangles, with great utility.
Also, to look again at Kempe’s “proof”: we can see where we can construct a vertex of degree 5 where his proof does not hold up. And we can try to turn that special case back into a map. The fact that nobody’s managed to construct an actual map relying on that flaw does not give any mathematical evidence that an example can’t exist. Staying within the field of math, the Bayesian is not updated and we can discard his conclusion. But we can step outside math’s rules and say “there’s a bunch of smart mathematicians trying to find a counterexample, and Kempe shows them exactly where the counterexample would have to be, and they can’t find one.” That fact updates the Bayesian, but reaches outside the field of math. The behavior of mathematicians faced by a math problem looks like part of mathematics, but actually isn’t.
Would you mind explain in a little more detail why you say a person who has seen Kempe’s flawed proof should have higher confidence than one who has not? Do you mean that it’s so emotionally compelling that one’s mind is convinced even if the math doesn’t add up? Or that the required (previously-hidden) premise that allows Kempe to ignore the degree 5 vertex has some possibility of truth, so that the conclusion has an increased likelihood of truth?
also: fixed the end.
Hmm, I’m not sure how to do so without just going through the whole proof. Essentially, Kempe’s proof showed that a smallest counterexample graph couldn’t have certain properties. One part of the proof was showing that the graph could not contain a vertex of degree 5. But this part was flawed. But Kempe did show that it couldn’t contain a vertex of degree 4, and moreover, it showed that any minimal counterexample must have a vertex of degree 5. This makes us more confident in the original claim since a minimal counterexample has to have a very restricted looking form.
Replying to the fixed end here so as to minimize confusion:
Well, yes but the claim I was addressing was that the claim you made that “encountering a single incorrect premise or step means that the conclusion has zero utility to the Bayesian” which is wrong. I agree that a flawed proof is not a proof.
And yes, the logic is in any case informal. See my earlier parenthetical remark. I actually consider the problem of confidence in mathematical reasoning to be one of the great difficult open problems within Bayesianism. One reason I don’t (generally) self-identify as a Bayesian is due to an apparent lack of this theory. (This itself deserves a disclaimer that I’m by no means at all an expert in this field and so there may be work in this direction but if so I haven’t seen any that is at all satisfactory.)
I think you are assuming I count a dubious premise as an incorrect premise. Obviously, a merely dubious premise allows the conclusion to have some utility to the Bayesian.
I really don’t think we actually disagree.
Really? Even incorrect premises can be useful. For example, one plausibility argument for the Riemann hypothesis rests on assuming that the Mobius function behaves like a random variable. But that’s a false statement. Nevertheless, it acts close enough to being a random variable that many find this argument to be evidence for RH. And there’s been very good work trying to take this false statement and make true versions of it.
Similarly, if one believes what you have said then one would have to conclude that if one lived in the 1700s that all of calculus would have been useless because it rests on the notion of infinitesimals which didn’t exist. The premise was incorrect, but the results were sound.
Incidentally, as more evidence, apparently this AC0 conjecture has just been proved true by Ben Green (rather, he noticed that other people had already done stuff that had this as a consequence, which the people asking the question hadn’t known about).
Ok, I need to refine my description of math a bit. I’d claimed that an incorrect premise gives useless conclusions; actually as you point out if we have a close-to-correct premise instead, we can have useful conclusions. The word “instead” is important there, because otherwise we can then add in a correct contradictory premise, generating new and false conclusions. In some sense this is necessary to all math, most evidently geometry: we don’t actually have any triangles in the world, but we use near-triangles all the time, pretending they’re triangles, with great utility.
Also, to look again at Kempe’s “proof”: we can see where we can construct a vertex of degree 5 where his proof does not hold up. And we can try to turn that special case back into a map. The fact that nobody’s managed to construct an actual map relying on that flaw does not give any mathematical evidence that an example can’t exist. Staying within the field of math, the Bayesian is not updated and we can discard his conclusion. But we can step outside math’s rules and say “there’s a bunch of smart mathematicians trying to find a counterexample, and Kempe shows them exactly where the counterexample would have to be, and they can’t find one.” That fact updates the Bayesian, but reaches outside the field of math. The behavior of mathematicians faced by a math problem looks like part of mathematics, but actually isn’t.