Even "artificial" methods are "discoveries"; "this method can be used to achieve that" is simply a consequence of the laws of nature, discovered by the inventors. An elevator may be seen as man-made, but the engineering of it is still a method, whose usefulness is merely a consequence of laws of nature, hence "discovered". Yet a patent does not "protect" a single elevator, but the method that makes it work. A star can't be patented, but neither can an elevator.
What you advocate is also against the spirit of patents, which is to protect science progress by providing financial incentives to make effort to contribute to science, whether or not some people would think such contribution should be be labelled a "discovery" or an "invention".
I don't have an answer as to whether we should patent mathematical methods, but your argument against it is very weak.
It goes both ways; there may be a lot of sub-par lazy professors out there, but the fruits are so valuable that you can see it as buying a lottery ticket that actually has a positive expected payout. It's hard to picture a world where breakthroughs on difficult problems and honest research that lead to controversial findings don't exist, so we get too used to it - and forget these can't be allowed to happen if they are hindered by poor academic culture and policies.
A lot of professors that work hard are protected that way. Old professors also do not have to directly compete with young researchers, so it's much more likely a committee would make sensible faculty hiring / tenure decisions that are free from conflict of interest. It's another issue that academics have to deal with other kinds of competition, such as research funding, as well as erosion of the tenure system by e.g. abusing adjunct professors.
There do exist professors who switched to less hype-driven research topics once they got tenure; for the most convicted, it's part of playing a game to get their messages through. As you can see it's already a hard game to play, so we don't really want to make it even harder. It's also easier (and quite common) for researchers to move on to more difficult topics after they get tenure, where doing it before tenure is close to self-destruction. Andrew Wiles used his tenure status to gain 7 years of solitude, slow publishing small pieces of research he accumulated previously, just enough to avoid getting fired, and proved Fermat's last theorem, a centuries-old open problem. And there are actually more of these people than you think, that are willing to play some games while working hard even when nobody's watching.
It's also not so clean cut that you can only do popular research to get tenure. Having a "good old boys club" can do bad and do good; a bad academic culture is harder to attack, while good academic culture is also more easily preserved. At a place that has a well-established healthy culture, it's easier to do `unpopular' research and still get tenured, and keep it that way. Sure these tend to happen in top institutions with an abundance of resources and strong attraction to top researchers, but it can be sustained partly because there are strong protection mechanisms within these institutions.
Does the tenure concept need to be refined? Probably. Does it bring good to all universities? Probably not. But it is one of the strongest foundations of a thriving academia, which far-reaching effects, like a bottom piece in Jenga, so you need a view of the big picture and bring a much more stronger argument before you can take it away.
Boris Karpichkov worked as a KGB agent in the 1980s before fleeing to Britain as a place of safety. He talks about his career, why Russian spies are again targeting Britain – and why he'll never stop looking over his shoulder
With the return to power of elements of the KGB, most notably Vladimir Putin, Kalugin was again accused of treason. In 2002 he was put on trial in absentia in Moscow and found guilty of spying for the West.[3] He was sentenced to fifteen years in jail,[6] in a verdict he described as "Soviet justice, which is really triumphant today".[7] The US and Russia have no extradition treaty.[7] Kalugin currently works for the Centre for Counterintelligence and Security Studies (CI CENTRE) is a member of the advisory board for the International Spy Museum.[8] He remains a critic of Vladimir Putin, a former subordinate, whom he called a "war criminal" over his conduct of the Second Chechen War.[3][9]
So there would soon be no more sweatshops for corporate America. Watch your economy crumble.
Krugman spouts nonsense at times, but this one is appalling. Pollution in China involves a lot of forces, including the `clean' countries, acting in their own interests and he can't possibly fail to understand that. Neither the Chinese government nor the US corporations would like such a change. The root of the problem is that some people like to earn money by messing up the world.
This proves he's just a propaganda mouthpiece, to help the US make a handsome profit from polluting activities around the world, while shedding every single bit of responsibility.
Sadly, these popular math 'geniuses' and child 'geniuses' never seem to do a damn thing that's truly notable.
Perhaps except Terrence Tao; a famous math prodigy, who also became an incredibly successful mathematician, "Such is Tao's reputation that mathematicians now compete to interest him in their problems, and he is becoming a kind of Mr Fix-it for frustrated researchers. "If you're stuck on a problem, then one way out is to interest Terence Tao," says Charles Fefferman [professor of mathematics at Princeton University].". Also Erik Demaine, who finished PhD and became a professor at MIT at 20; he has a less impressive history than Tao, but still a fruitful career.
Top Ten Things Overheard At The ANSI C Draft Committee Meetings: (5) All right, who's the wiseguy who stuck this trigraph stuff in here?