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Understanding is the new bottleneck
Understanding limits the pace
Having AI produce a solution doesn't mean you understand it, and it doesn't give you an idea of what other problems you'd like solved. People will be rate limited by how fast they can understand plausibly true solutions. Science has always been about working out what is true and what is not, and that job hasn't changed.
A powerful new tool
Mathematicians now have a powerful new tool, like miners getting hydraulic excavators or powered drills. The amount of math out there to be discovered is probably infinite, so using the tool takes a lot of work.
Harder problems ahead
The work stays prestigious too, because most people can't do it. Even if any high school kid can press a button and solve the Riemann Hypothesis, the output will be gibberish to them. Any given problem gets easier, since understanding a solution is generally easier than finding one. AI will also produce much harder problems than any made so far, some so difficult that understanding the solution is as hard for a human as solving the Jacobian Conjecture without a machine.
Why is checking an answer easier than finding one?
Computer scientists have a formal version of this question, called P versus NP. A completed sudoku grid can be checked in moments, yet finding the solution to a hard puzzle can take far longer. In a 2019 poll by William Gasarch, about 88 percent of surveyed experts believed the two are not the same, though nobody has proven it. The Clay Mathematics Institute lists P versus NP among its seven Millennium Prize Problems, each carrying a one million dollar prize.
Why has the Jacobian Conjecture resisted proof?
Ott-Heinrich Keller posed the conjecture in 1939. It concerns polynomial maps, and it says that if a certain determinant built from the map's derivatives is a nonzero constant, then the map has a polynomial inverse. It is easy to confirm for a single variable and has been open in higher dimensions. In 1982, Hyman Bass, Edwin Connell, and David Wright showed that proving it for maps of degree three in every dimension would settle all cases.
How did powered drills change mining and tunneling?
Hand drilling of rock was slow, and crews of miners swung hammers against steel bits for hours. In the 1860s, compressed-air rock drills began to replace that labor in mines and tunnels. The Hoosac Tunnel in Massachusetts was among the first projects in the United States to use compressed-air drills and nitroglycerin. It was completed in 1875.
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