Could you elaborate on the role of the 25k mathematics papers on the arXiv in leading you to that 99.9999% figure? I’m having trouble following the logic in that paragraph.
I was hinting at the sheer volume of mathematical research — given that there’s been so much mathematical research, and the fact that I haven’t heard of any examples of statements as robustly supported as the product formula for sine that turned out to be false, there should be a very strong prior against such a statement being false.
One could attribute my not having heard of such an example to my ignorance :-) but such a story would have a strong tendency to percolate on account of being so weird, so one wouldn’t need to know a great deal to have heard it.
I understand how that works qualitatively, but I don’t understand where the number 99.9999% comes from, nor how the number 25000 feeds into it. And surely only a tiny fraction of mathematics papers on the arXiv deal with conjectures of this sort, so why cite the number of papers there in particular?
I’ll be (rather bogusly) quantitative for a moment. Pretend that every single one of those 25k papers on the arXiv makes an argument similar to Euler’s, and that if any one of them were wrong then you’d certainly have heard about it. How improbable would an error have to be, to make it unsurprising (say, p=1/2) that you haven’t heard of one? Answer: the probability of one paper being correct would have to be at least about 99.997%.
Now, to be sure, there’s more out there than the arXiv. But, equally, hardly any papers deal with arguments like Euler’s, and many papers go unscrutinized and could be wrong without anyone noticing, and surely many are obscure enough that even if they were noticed the news might not spread.
Maybe I’m being too pernickety. But it seems to me that one oughtn’t to say “In the context of the fact that ~25,000 math papers were posted on ArXiv in 2012 it may be reasonable to conclude that the appropriate confidence level would have been 99.9999+%” when one means “There are lots of mathematics papers published and I haven’t heard of another case of something like this being wrong, so probably most such cases are right”.
I agree that my argument isn’t tight. I’m partially going on tacit knowledge that I acquired during graduate school. The figure that I gave is a best guess.
I changed my post accordingly. I’m somewhat puzzled as to why it rubbed people the wrong way (to such a degree that my above comment was downvoted three times.)
Could you elaborate on the role of the 25k mathematics papers on the arXiv in leading you to that 99.9999% figure? I’m having trouble following the logic in that paragraph.
I was hinting at the sheer volume of mathematical research — given that there’s been so much mathematical research, and the fact that I haven’t heard of any examples of statements as robustly supported as the product formula for sine that turned out to be false, there should be a very strong prior against such a statement being false.
One could attribute my not having heard of such an example to my ignorance :-) but such a story would have a strong tendency to percolate on account of being so weird, so one wouldn’t need to know a great deal to have heard it.
I understand how that works qualitatively, but I don’t understand where the number 99.9999% comes from, nor how the number 25000 feeds into it. And surely only a tiny fraction of mathematics papers on the arXiv deal with conjectures of this sort, so why cite the number of papers there in particular?
I’ll be (rather bogusly) quantitative for a moment. Pretend that every single one of those 25k papers on the arXiv makes an argument similar to Euler’s, and that if any one of them were wrong then you’d certainly have heard about it. How improbable would an error have to be, to make it unsurprising (say, p=1/2) that you haven’t heard of one? Answer: the probability of one paper being correct would have to be at least about 99.997%.
Now, to be sure, there’s more out there than the arXiv. But, equally, hardly any papers deal with arguments like Euler’s, and many papers go unscrutinized and could be wrong without anyone noticing, and surely many are obscure enough that even if they were noticed the news might not spread.
Maybe I’m being too pernickety. But it seems to me that one oughtn’t to say “In the context of the fact that ~25,000 math papers were posted on ArXiv in 2012 it may be reasonable to conclude that the appropriate confidence level would have been 99.9999+%” when one means “There are lots of mathematics papers published and I haven’t heard of another case of something like this being wrong, so probably most such cases are right”.
I agree that my argument isn’t tight. I’m partially going on tacit knowledge that I acquired during graduate school. The figure that I gave is a best guess.
I think it’s inappropriate to cite a figure as support for your estimate unless you indicate in some way how that figure affects your estimate.
I changed my post accordingly. I’m somewhat puzzled as to why it rubbed people the wrong way (to such a degree that my above comment was downvoted three times.)