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In an Atlantic essay, “Why The Past Ten Years of American Life have Been Uniquely Stupid,” and in a follow-up piece, Haidt argued that the Age of Gutenberg–of books and the depth understanding that comes with them–ended somewhere around 2014 with the rise of “Share,” “Like” and “Retweet” buttons that opened the way for trolls, hucksters and Trumpists

The new age of “hyper-virality,” he writes, has given us both January 6 and cancel culture–ugly polarization in both directions. On the subject of stupidification, we should add the fact that high school students now get virtually their entire stock of knowledge about the world from digital platforms.

Haidt proposed several reforms, including modifying Facebook’s “Share” function and requiring “user verification” to get rid of trolls. But he doesn’t really believe in his own medicine

Haidt said that the era of “shared understanding” is over–forever. When I asked if he could envision changes that would help protect democracy, Haidt quoted Goldfinger: “Do you expect me to talk?” “No, Mr. Bond, I expect you to die!”

Social media is a public health hazard–the cognitive equivalent of tobacco and sugary drinks. Adopting a public health model, we could, for examople, ban the use of algorithms to reduce virality, or even require social media platforms to adopt a subscription rather than advertising revenue model and thus remove their incentive to amass ev er more eyeballs.

We could, but we won’t, because unlike other public health hazards, digital platforms are forms of speech. Fox New is probably responsible for more polarization than all social media put together, but the federal government could not compel it–and all other media firms–to change its revenue model.

If Mark Zuckerberg or Elon Musk won’t do so out of concern for the public good–a pretty safe bet–they could be compelled to do so only by public or competitive pressure. 

Taiwan has provide resilient because its society is resilient; people reject China’s lies. We, here, don’t lack for fact-checkers, but rather for people willing to believe them. The problem is not the technology, but ourselves.

you have to wonder if people really are repelled by our poisonous discourse, or by the hailstorm of disinformation, or if they just want to live comfortably inside their own bubble, and not somebody else’

If Jonathan Haidt is right, it’s not because we’ve created a self-replicating machine that is destined to annihilate reason; it’s because we are the self-replicating machine.

The human problem-solvers — 609 high school students from 108 countries — won 58 gold medals, 123 silver and 145 bronze. The A.I. performed at the level of a silver medalist, solving four out of six problems for a total of 28 points. It was the first time that A.I. has achieved a medal-worthy performance on an Olympiad’s problems.

Nonetheless, Dr. Kohli described the result as a “phase transition” — a transformative change — “in the use of A.I. in mathematics and the ability of A.I. systems to do mathematics.”

Dr. Gowers added in an email: “I was definitely impressed.” The lab had discussed its Olympiad ambitions with him a couple of weeks beforehand, so “my expectations were quite high,” he said. “But the program met them, and in one or two instances significantly surpassed them.” The program found the “magic keys” that unlocked the problems, he said.

Haojia Shi, a student from China, ranked No. 1 and was the only competitor to earn a perfect score — 42 points for six problems; each problem is worth seven points for a full solution. The U.S. team won first place with 192 points; China placed second with 190.

The Google system earned its 28 points for fully solving four problems — two in algebra, one in geometry and one in number theory. (It flopped at two combinatorics problems.) The system was allowed unlimited time; for some problems it took up to three days. The students were allotted only 4.5 hours per exam.

“The fact that we’ve reached this threshold, where it’s even possible to tackle these problems at all, is what represents a step-change in the history of mathematics,” he added. “And hopefully it’s not just a step-change in the I.M.O., but also represents the point at which we went from computers only being able to prove very, very simple things toward computers being able to prove things that humans can’t.”

“Mathematics requires this interesting combination of abstract, precise and creative reasoning,” Dr. Davies said. In part, he noted, this repertoire of abilities is what makes math a good litmus test for the ultimate goal: reaching so-called artificial general intelligence, or A.G.I., a system with capabilities ranging from emerging to competent to virtuoso to superhuman

One approach was an informal reasoning system, expressed in natural language. This system leveraged Gemini, Google’s large language model. It used the English corpus of published problems and proofs and the like as training data.

Dr. Hubert’s team developed a new model that is comparable but more generalized. Named AlphaProof, it is designed to engage with a broad range of mathematical subjects. All told, AlphaGeometry and AlphaProof made use of a number of different A.I. technologies.

In January, a Google DeepMind system named AlphaGeometry solved a sampling of Olympiad geometry problems at nearly the level of a human gold medalist. “AlphaGeometry 2 has now surpassed the gold medalists in solving I.M.O. problems,” Thang Luong, the principal investigator, said in an email.

The informal system excels at identifying patterns and suggesting what comes next; it is creative and talks about ideas in an understandable way. Of course, large language models are inclined to make things up — which may (or may not) fly for poetry and definitely not for math. But in this context, the L.L.M. seems to have displayed restraint; it wasn’t immune to hallucination, but the frequency was reduced.

Another approach was a formal reasoning system, based on logic and expressed in code. It used theorem prover and proof-assistant software called Lean, which guarantees that if the system says a proof is correct, then it is indeed correct. “We can exactly check that the proof is correct or not,” Dr. Hubert said. “Every step is guaranteed to be logically sound.”

Another crucial component was a reinforcement learning algorithm in the AlphaGo and AlphaZero lineage. This type of A.I. learns by itself and can scale indefinitely, said Dr. Silver, who is Google DeepMind’s vice-president of reinforcement learning. Since the algorithm doesn’t require a human teacher, it can “learn and keep learning and keep learning until ultimately it can solve the hardest problems that humans can solve,” he said. “And then maybe even one day go beyond those.”

Dr. Hubert added, “The system can rediscover knowledge for itself.” That’s what happened with AlphaZero: It started with zero knowledge, Dr. Hubert said, “and by just playing games, and seeing who wins and who loses, it could rediscover all the knowledge of chess. It took us less than a day to rediscover all the knowledge of chess, and about a week to rediscover all the knowledge of Go. So we thought, Let’s apply this to mathematics.”

Dr. Gowers doesn’t worry — too much — about the long-term consequences. “It is possible to imagine a state of affairs where mathematicians are basically left with nothing to do,” he said. “That would be the case if computers became better, and far faster, at everything that mathematicians currently do.”

“There still seems to be quite a long way to go before computers will be able to do research-level mathematics,” he added. “It’s a fairly safe bet that if Google DeepMind can solve at least some hard I.M.O. problems, then a useful research tool can’t be all that far away.”

A really adept tool might make mathematics accessible to more people, speed up the research process, nudge mathematicians outside the box. Eventually it might even pose novel ideas that resonate.