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New financial constraints on family formation are a potential cause, as implied by another striking finding in the Journal poll — 78 percent of adults lack confidence this generation of children will enjoy a better life than they do.

Yet a recent analysis for the Aspen Economic Strategy Group by Melissa S. Kearney and Phillip B. Levine, economics professors at the University of Maryland and Wellesley College, respectively, determined that “beyond the temporary effects of the Great Recession, no recent economic or policy change is responsible for a meaningful share of the decline in the US fertility rate since 2007.”

Their study took account of such factors as the high cost of child care, student debt service and housing as well as Medicaid coverage and the wider availability of long-acting reversible contraception. Yet they had “no success finding evidence” that any of these were decisive.

Kearney and Levine speculated instead that the answers lie in the cultural zeitgeist — “shifting priorities across cohorts of young adults,”

the timing and the magnitude of such a demographic sea-change cry out for explanation. What happened in 2007?

another event that year: Apple released the first iPhone, a revolutionary cultural moment if there ever was one. The ensuing smartphone-enabled social media boom — Facebook had opened membership to anyone older than 13 in 2006 — forever changed how human beings relate with one another.

We are just beginning to understand this development’s effect on mental health, education, religious observance, community cohesion — everything. Why wouldn’t it also affect people’s willingness to have children?

one indirect way new media affect childbearing rates is through “time competition effects” — essentially, hours spent watching the tube cannot be spent forming romantic partnerships.

a 2021 review of survey data on young adults and adolescents in the United States and other countries, the years between 2009 and 2018 saw a marked decline in reported sexual activity.

the authors hypothesized that people are distracted from the search for partners by “increasing use of computer games and social media.

during the late 20th century, Brazil’s fertility rates fell after women who watched soap operas depicting smaller families sought to emulate them by having fewer children themselves.

This may be an area where incentives do not influence behavior, at least not enough. Whether the cultural shift to lower birthrates occurs on an accelerated basis, as in the United States after 2007, or gradually, as it did in Japan, it appears permanent — “sticky,” as policy wonks say.

While people praised Finland’s strong social safety net and spoke glowingly of the psychological benefits of nature and the personal joys of sports or music, they also talked about guilt, anxiety and loneliness. Rather than “happy,” they were more likely to characterize Finns as “quite gloomy,” “a little moody” or not given to unnecessary smiling

Many also shared concerns about threats to their way of life, including possible gains by a far-right party in the country’s elections in April, the war in Ukraine and a tense relationship with Russia, which could worsen now that Finland is set to join NATO.

It turns out even the happiest people in the world aren’t that happy. But they are something more like content.

Finns derive satisfaction from leading sustainable lives and perceive financial success as being able to identify and meet basic need

“In other words,” he wrote in an email, “when you know what is enough, you are happy.”

“‘Happiness,’ sometimes it’s a light word and used like it’s only a smile on a face,” Teemu Kiiski, the chief executive of Finnish Design Shop, said. “But I think that this Nordic happiness is something more foundational.”

e conventional wisdom is that it’s easier to be happy in a country like Finland where the government ensures a secure foundation on which to build a fulfilling life and a promising future. But that expectation can also create pressure to live up to the national reputation.

“We are very privileged and we know our privilege,” said Clara Paasimaki, 19, one of Ms. Hansen’s students in Kokkola, “so we are also scared to say that we are discontent with anything, because we know that we have it so much better than other people,” especially in non-Nordic countries.

“The fact that Finland has been ‘the happiest country on earth’ for six years in a row could start building pressure on people,” he wrote in an email. “If we Finns are all so happy, why am I not happy?

Since immigrating from Zimbabwe in 1992, Julia Wilson-Hangasmaa, 59, has come to appreciate the freedom Finland affords people to pursue their dreams without worrying about meeting basic needs

“Back in the day when it wasn’t that easy to survive the winter, people had to struggle, and then it’s kind of been passed along the generations,” said Ms. Paasimaki’s classmate Matias From, 18. “Our parents were this way. Our grandparents were this way. Tough and not worrying about everything. Just living life.”

The Finnish way of life is summed up in “sisu,” a trait said to be part of the national character. The word roughly translates to “grim determination in the face of hardships,” such as the country’s long winters: Even in adversity, a Finn is expected to persevere, without complaining.

When she returns to her home country, she is struck by the “good energy” that comes not from the satisfaction of sisu but from exuberant joy.

“What I miss the most, I realize when I enter Zimbabwe, are the smiles,” she said, among “those people who don’t have much, compared to Western standards, but who are rich in spirit.”

Many of our subjects cited the abundance of nature as crucial to Finnish happiness: Nearly 75 percent of Finland is covered by forest, and all of it is open to everyone thanks to a law known as “jokamiehen oikeudet,” or “everyman’s right,” that entitles people to roam freely throughout any natural areas, on public or privately owned land.

“I enjoy the peace and movement in nature,” said Helina Marjamaa, 66, a former track athlete who represented the country at the 1980 and 1984 Olympic Games. “That’s where I get strength. Birds are singing, snow is melting, and nature is coming to life. It’s just incredibly beautiful.”

“I am worried with this level of ignorance we have toward our own environment,” he said, citing endangered species and climate change. The threat, he said, “still doesn’t seem to shift the political thinking.”

Born 17 years after Finland won independence from Russia, Eeva Valtonen has watched her homeland transform: from the devastation of World War II through years of rebuilding to a nation held up as an exemplar to the world.

“My mother used to say, ‘Remember, the blessing in life is in work, and every work you do, do it well,’” Ms. Valtonen, 88, said. “I think Finnish people have been very much the same way. Everybody did everything together and helped each other.”

Maybe it isn’t that Finns are so much happier than everyone else. Maybe it’s that their expectations for contentment are more reasonable, and if they aren’t met, in the spirit of sisu, they persevere.

“We don’t whine,” Ms. Eerikainen said. “We just do.”

Before the introduction of Botox and hyaluronic acid dermal fillers in 2002 and 2003, respectively, aesthetic work was serious, expensive. Nose jobs and face lifts required general anesthesia, not insignificant recovery time, and cost thousands of dollars (in 2000, a facelift was $5,416 on average, and a rhinoplasty $4,109, around $9,400 and $7,000 adjusted).

In contrast, the average price of a syringe of hyaluronic acid filler today is $684, while treating, for example, the forehead and eyes with Botox will put you out anywhere from $300 to $600

Botox and filler only accelerated a trend that began in the ’70s and ’80s and is just now reaching its saturation point.

In 2018, use of Botox and fillers was up 18% and 20% from five years prior. Philosophies of prejuvenation have made Botox use jump 22% among 22- to 37-year-olds in half a decade as well. By 2030, global noninvasive aesthetic treatments are predicted to triple.

The trouble is that a status symbol, without status, is common.

Beauty has always been exclusive. When someone strikes you as pretty, it means they are something that everyone else is not.

It’s a zero-sum game, as relative as our morals. Naturally, we hoard of beauty what we can. It’s why we call grooming tips “secrets.”

Largely the secrets started with the wealthy, who possess the requisite money and leisure to spare on their appearances

We copied the beautiful and the rich, not in facsimile, but in homage.

we didn’t have the tools for anything more than emulation. Fake breasts and overdrawn lips only approximated real ones; a birthmark drawn with pencil would always be just that.

Instagram Face, on the other hand, distinguishes itself by its sheer reproducibility. Not only because of those new cosmetic technologies, which can truly reshape features, at reasonable cost and with little risk.

built in to the whole premise of reversible, low-stakes modification is an indefinite flux, and thus a lack of discretion.

Instagram Face has replicated outward, with trendsetters giving up competing with one another in favor of looking eerily alike. And obviously it has replicated down.

But the more rapidly it replicates, and the clearer our manuals for quick imitation become, the closer we get to singularity—that moment Kim Kardashian fears unlike any other: the moment when it becomes unclear whether we’re copying her, or whether she is copying us.

natural looks have always been, and still are, more valuable than artificial ones. Partly because of our urge to legitimize in any way we can the advantages we have over other people. Hotness is a class struggle.

As more and more women post videos of themselves eating, sleeping, dressing, dancing, and Only-Fanning online, in a logical bid for economic ascendance, the women who haven’t needed to do that gain a new status symbol.

Privacy. A life which is not a ticketed show. An intimacy that does not admit advertisers. A face that does not broadcast its insecurity, or the work undergone to correct it.

Upper class, private women get discrete work done. The differences aren’t in the procedures themselves—they’re the same—but in disposition

Eva, who lives between central London, Geneva, and the south of France, says: “I do stuff, but none of the stuff I do is at all in my head associated with Instagram Face. Essentially you do similar procedures, but the end goal is completely different. Because they are trying to get the result of looking like another human being, and I’m just beautifying myself.”

Eva looks like Eva. If she has procedures in common with Kim K, you couldn’t tell. “I look at my features and I think long and hard of how I can, without looking different and while keeping as natural as possible, make them look better and more proportional. I’m against everything that is too invasive. My problem with Instagram Face is that if you want to look like someone else, you should be in therapy.”

what he restores is complicated and yet not complicated at all. It’s herself, the fingerprint of her features. Her aura, her presence and genealogy, her authenticity in space and time.

Dr. Taktouk’s approach is “not so formulaic.” He aims to give his patients the “better versions of themselves.” “It’s not about trying to be anyone else,” he says, “or creating a conveyor belt of patients. It’s about working with your best features, enhancing them, but still looking like you.”

“Vulgar” says that in pursuing indistinguishability, women have been duped into another punishing divide. “Vulgar” says that the subtlety of his work is what signals its special class—and that the women who’ve obtained Instagram Face for mobility’s sake have unwittingly shut themselves out of it.

While younger women are dissolving their gratuitous work, the 64-year-old Madonna appeared at the Grammy Awards in early February, looking so tragically unlike herself that the internet launched an immediate postmortem.

The folly of Instagram Face is that in pursuing a bionic ideal, it turns cosmetic technology away from not just the reality of class and power, but also the great, poignant, painful human project of trying to reverse time. It misses the point of what we find beautiful: that which is ephemeral, and can’t be reproduced

Age is just one of the hierarchies Instagram Face can’t topple, in the history of women striving versus the women already arrived. What exactly have they arrived at?

Youth, temporarily. Wealth. Emotional security. Privacy. Personal choices, like cosmetic decisions, which are not so public, and do not have to be defended as empowered, in the defeatist humiliation of our times

Maybe they’ve arrived at love, which for women has never been separate from the things I’ve already mentioned.

I can’t help but recall the time I was chatting with a plastic surgeon. I began to point to my features, my flaws. I asked her, “What would you do to me, if I were your patient?” I had many ideas. She gazed at me, and then noticed my ring. “Nothing,” she said. “You’re already married.”

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

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.

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.

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.

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.