We have an approximate expression of how many prime numbers there are less than N, as N gets larger. If the Riemann hypothesis is true, then we know that the errors in this approximation are nice and small, which would allow us to prove many other approximate results. (There are many results of the form "If the Riemann hypothesis is true, then...")
It probably doesn't open up a path to a full proof, simply because Guth and Maynard are excellent mathematicians, and they wouldn't have given away their approach yet if they thought they could prove the full thing.
It's still important because there hasn't been much new evidence in years whether the conjecture is true or false. Now we have some new evidence that it's true.
Allen is probably the leading economic historian of the Industrial Revolution. Reading someone want to dismiss his book because someone used the same painting on their book cover later is probably the most HN comment I've ever seen.
Mathematicians could do a better job of conveying their topics, but this is really not the case. Some things are just genuinely hard, and the research frontier in mathematics is one of these. If you could produce a 5-page proof of an important theorem that currently has a 100-page proof, you would be hailed as a hero.
It's not art snobs who made it famous, either. It's just famous for being famous, at that poin. In the same room at the Louvre there's Veronese's "The Wedding at Cana" and several Titians that art snobs would also linger over, but the room is 90% tourists who want to see the Mona Lisa.
While a part of the standard chess rules for over a century, it's common for people who learn to play with their family as kids don't learn it. I didn't learn it, either, and only found out about it when I started playing chess programs online.
This is common enough that "Google en passant" is a meme in some circles.
I was wondering this. How often does it turn out that a drug has no serious downsides? I guess penicillin was, if you're not allergic. (There is microbes evolving antibiotic resistance, but that's probably not a concern for ozempic.)
I don't have a clear view on whether LLMs are intelligent or not, but I don't understand your argument at all. Why couldn't an intelligent agent exist that refuses to admit they don't know something, and just make stuff up? I've known people who come pretty close to this behavior.
The influence of D&D on pop culture in Japan is so readily visible (see Delicious in Dungeon on Netflix right now) that it's surprising to read that its commercial success was so fleeting.
Minsky is an example of the power of credentialism. He was an MIT professor who researched AI. I think history has pretty much demonstrated he didn't have any special insight into the question of AI, but for decades he was the most famous researcher in the field.
There's got to be some miscommunication between the researchers and the journalists as to the contribution. I know anthropology undergraduates who years ago had whole courses where they had to make and use their own stone tools.
If it doesn't really have a fixed point you get a function that never terminates. In denotational semantics you define a special value, "bottom" that means "undefined", and the fixed point is bottom.
And it's not all functions, just computable ones. But if there's algorithm to compute non-trivial zeroes of the zeta function, it would just never terminate if there aren't any.
I took your advice, and you're right, it is very readable. Seeing it in its original context also helps me understand why a logician would want to invent it.