How does the distribution of truth compare to the distribution of opinion? That is, consider some spectrum of possible answers, like the point difference in a game, or the sea level rise in the next century. On each such spectrum we could get a distribution of (point-estimate) opinions, and in the end a truth. So in each such case we could ask for truth’s opinion-rank: what fraction of opinions were less than the truth? For example, if 30% of estimates were below the truth (and 70% above), the opinion-rank of truth was 30%.
"I am not aware of any story of disappointment with a proof of a conjecture that did not involve incomprehension of the proof"
De Branges' proof of the Bieberbach conjecture left some people cold because it rested on a magic trick taken out of a hat.
"I am not aware of anyone who groaned at Perelman's solution, except maybe Yau"
Why should Yau be upset? It was Yau's approach that Perelman pushed through. The folks that groaned were the topologists: here is the conjecture that has made their field being solved by PDE methods, no wonder they were not pleased. (By the way, I'm not trying to put down Perelman or anybody else, but if we don't discuss specific examples it's hard to make any progress).
"Your claim that people would be disappointed in a spectral proof of the Riemann hypothesis is far more absurd".
I don't understand why you think it's absurd.
Finally, I'm too ignorant about P=NP to be able to say anything worthwhile, so I'll shut up on that subject.
PS: This article in today's arxiv (by the recent fields medalist Tao) gives a good description of the various different truths in mathematics and how they come about (quite uexpectedly I would say).
Before I go off on Calca's tangent, complaining about his examples, I will give an example on the difficulty of observing conjectures that are deeply held. One does know that everyone was shocked by IP=PSPACE, but one can only tell that people conjectured a strict inclusion by the difficulty in publishing the inclusion of IP in PSPACE. Of course, I do know this, and I know it because the theorem is celebrated, and that is because, per Calca, it was entirely unexpected and changed the perspective. And they tell the story to demonstrate this. Was the insolubility by radicals of the quintic shocking 200 years ago? I have no idea.
It is true that mathematicians care about how conjectures are proved, and what those methods give more generally. This is the main complaint about extremely computer-aided proofs, like the four color theorem. There is the story of Grothendieck's disappointment with Deligne's proof of the Weil conjectures. Aside from that joke, I am not aware of any story of disappointment with a proof of a conjecture that did not involve incomprehension of the proof (which happens not too rarely with proofs by humans).
I am not aware of anyone who groaned at Perelman's solution, except maybe Yau. Perelman didn't introduce any radically new concepts, but he showed that the Gromov-Hausdorff degeneration techniques are very useful in the context of Ricci flow. It is expected that this, and his other improvements to Ricci flow, will have application to Ricci flow in higher dimensions, particularly on Kaehler manifolds. I suspect that with equal distance from Wiles, you would describe his work as merely pushing through an established program.
Your claim that people would be disappointed in a spectral proof of the Riemann hypothesis is far more absurd. My impression is that the spectral approach is more an idle remark than an established program. Connes does have a specific operator (although I'm not sure it quite fits in the Hilbert-Polya picture) and I imagine that his conjectures about it would have other consequences than the Riemann hypothesis. Certainly, he claims that his perspective has other applications to number theory, but I don't know how broadly one must interpret "perspective."
The point about mathematical truth is that we value it differently whether it was expected or unexpected. Authors often make a point of framing their articles to make it sound like their results are totally unexpected (and say that they are even surprised themselves!). In reality, oftentimes, the reason the result looks unexpected is because the author is using some analogy or guideline of a different nature, say geometric, or algebraic, or physical, and is thus able to push some computations through that would otherwise be difficult to foresee. Take for instance the Riemann Hypothesis, it's probably true, but what people really care about is how it will be proved if true. If someone tomorrow proves it by showing that the zeroes are the spectrum of a self-adjoint operator, people will groan, just like some people have groaned at the Perelman solution of the Poincare' conjecture (because it just pushed through a well established program). People were much more excited about Wiles solution to Fermat's theorem, because of all the novel concepts he introduced. So the culture pushes people to explore the edges, not the mainstream, and even NSF seems to value "transformative", or "interdisciplinary" mathematics more than, say, traditional fields. But asking where truth lies with respect to opinions is a difficult question because the opinions themselves shape what is to be considered "interesting" truth.
Calca: It's clearly the case that proofs of results widely believed to be true are sometimes considered very valuable or important. Empirical historical result on the calibration of mathematical conjectures would be VERY valuable. Do you have any data along those lines? I would very much like to know whether, for instance, there is actually a .1% chance, a 1% chance, or a 10% chance that P = NP .
There's a well-known paper on how historical estimates of physical constants measured up against the modern values. IIRC, generally in those cases the truth was in the tails.
In Math the extremists seem to win out. Math is full of unexpected results, amazing connections and links between unrelated objects. In fact, if a middle of the road prediction (say a conjecture that is widely believed to hold true) is proved to be true then the result is usually regarded as not very significant.
I think that slogan applies to one of the measurable things you mention, sports results. I've spent some time analyzing sports betting by differences between theoretical strengths of teams, posted odds, and bettor's preferences based on which way the odds move after they're posted. There are a number of ways that such numbers point to bettor's favoring boring results, while actual results are more varied.
One of the most consistent features of baseball betting is that the posted odds of heavy home favorites are shorter than they should be according to actual results given the calculated strengths of the teams, maybe 3-1 instead of 5-1. I would think that represents a bias in bettors who start with the idea that either team can win rather than the reality that the home team will win 5 times out of 6, but the other issue is who wants to bet on a heavy favorite and make only 33 cents for a dollar vs. betting on the underdog and getting 3 dollars for a dollar if you get lucky? That would be even worse if the odds were where the actual results say they should be. I don't know which factor is more important, perception of the game or perception of the payoff.
I suspect there are other contexts where people vary more than reality, but I can't think of one right now. I'm not sure which slogan describes that, maybe something about regression to the mean, as people believe in things happening in streaks instead of in boring ways sometimes.
Douglas Knight,
"I am not aware of any story of disappointment with a proof of a conjecture that did not involve incomprehension of the proof"
De Branges' proof of the Bieberbach conjecture left some people cold because it rested on a magic trick taken out of a hat.
"I am not aware of anyone who groaned at Perelman's solution, except maybe Yau"
Why should Yau be upset? It was Yau's approach that Perelman pushed through. The folks that groaned were the topologists: here is the conjecture that has made their field being solved by PDE methods, no wonder they were not pleased. (By the way, I'm not trying to put down Perelman or anybody else, but if we don't discuss specific examples it's hard to make any progress).
"Your claim that people would be disappointed in a spectral proof of the Riemann hypothesis is far more absurd".
I don't understand why you think it's absurd.
Finally, I'm too ignorant about P=NP to be able to say anything worthwhile, so I'll shut up on that subject.
PS: This article in today's arxiv (by the recent fields medalist Tao) gives a good description of the various different truths in mathematics and how they come about (quite uexpectedly I would say).
http://www.arxiv.org/abs/ma...
Before I go off on Calca's tangent, complaining about his examples, I will give an example on the difficulty of observing conjectures that are deeply held. One does know that everyone was shocked by IP=PSPACE, but one can only tell that people conjectured a strict inclusion by the difficulty in publishing the inclusion of IP in PSPACE. Of course, I do know this, and I know it because the theorem is celebrated, and that is because, per Calca, it was entirely unexpected and changed the perspective. And they tell the story to demonstrate this. Was the insolubility by radicals of the quintic shocking 200 years ago? I have no idea.
It is true that mathematicians care about how conjectures are proved, and what those methods give more generally. This is the main complaint about extremely computer-aided proofs, like the four color theorem. There is the story of Grothendieck's disappointment with Deligne's proof of the Weil conjectures. Aside from that joke, I am not aware of any story of disappointment with a proof of a conjecture that did not involve incomprehension of the proof (which happens not too rarely with proofs by humans).
I am not aware of anyone who groaned at Perelman's solution, except maybe Yau. Perelman didn't introduce any radically new concepts, but he showed that the Gromov-Hausdorff degeneration techniques are very useful in the context of Ricci flow. It is expected that this, and his other improvements to Ricci flow, will have application to Ricci flow in higher dimensions, particularly on Kaehler manifolds. I suspect that with equal distance from Wiles, you would describe his work as merely pushing through an established program.
Your claim that people would be disappointed in a spectral proof of the Riemann hypothesis is far more absurd. My impression is that the spectral approach is more an idle remark than an established program. Connes does have a specific operator (although I'm not sure it quite fits in the Hilbert-Polya picture) and I imagine that his conjectures about it would have other consequences than the Riemann hypothesis. Certainly, he claims that his perspective has other applications to number theory, but I don't know how broadly one must interpret "perspective."
I think Tetlock has the best data I've seen on this.
Michael,
The point about mathematical truth is that we value it differently whether it was expected or unexpected. Authors often make a point of framing their articles to make it sound like their results are totally unexpected (and say that they are even surprised themselves!). In reality, oftentimes, the reason the result looks unexpected is because the author is using some analogy or guideline of a different nature, say geometric, or algebraic, or physical, and is thus able to push some computations through that would otherwise be difficult to foresee. Take for instance the Riemann Hypothesis, it's probably true, but what people really care about is how it will be proved if true. If someone tomorrow proves it by showing that the zeroes are the spectrum of a self-adjoint operator, people will groan, just like some people have groaned at the Perelman solution of the Poincare' conjecture (because it just pushed through a well established program). People were much more excited about Wiles solution to Fermat's theorem, because of all the novel concepts he introduced. So the culture pushes people to explore the edges, not the mainstream, and even NSF seems to value "transformative", or "interdisciplinary" mathematics more than, say, traditional fields. But asking where truth lies with respect to opinions is a difficult question because the opinions themselves shape what is to be considered "interesting" truth.
Calca: It's clearly the case that proofs of results widely believed to be true are sometimes considered very valuable or important. Empirical historical result on the calibration of mathematical conjectures would be VERY valuable. Do you have any data along those lines? I would very much like to know whether, for instance, there is actually a .1% chance, a 1% chance, or a 10% chance that P = NP .
James, I had a post on that paper on Dec 26. But examples of overconfident error estimates is not quite what I'm looking for.
DavidD, the standard example of the long-shot bias does suggest the truth is in the hump at the track.
There's a well-known paper on how historical estimates of physical constants measured up against the modern values. IIRC, generally in those cases the truth was in the tails.
In Math the extremists seem to win out. Math is full of unexpected results, amazing connections and links between unrelated objects. In fact, if a middle of the road prediction (say a conjecture that is widely believed to hold true) is proved to be true then the result is usually regarded as not very significant.
Truth is stranger than fiction.
I think that slogan applies to one of the measurable things you mention, sports results. I've spent some time analyzing sports betting by differences between theoretical strengths of teams, posted odds, and bettor's preferences based on which way the odds move after they're posted. There are a number of ways that such numbers point to bettor's favoring boring results, while actual results are more varied.
One of the most consistent features of baseball betting is that the posted odds of heavy home favorites are shorter than they should be according to actual results given the calculated strengths of the teams, maybe 3-1 instead of 5-1. I would think that represents a bias in bettors who start with the idea that either team can win rather than the reality that the home team will win 5 times out of 6, but the other issue is who wants to bet on a heavy favorite and make only 33 cents for a dollar vs. betting on the underdog and getting 3 dollars for a dollar if you get lucky? That would be even worse if the odds were where the actual results say they should be. I don't know which factor is more important, perception of the game or perception of the payoff.
I suspect there are other contexts where people vary more than reality, but I can't think of one right now. I'm not sure which slogan describes that, maybe something about regression to the mean, as people believe in things happening in streaks instead of in boring ways sometimes.
Will, a dataset of football score forecasts and final scores should be relevant.
I doubt much data exists on this. The truth is an ongoing debate.