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An aperiodic tiling displays no such “translational symmetry,” and mathematicians have long sought a single shape that could tile the plane in such a fashion. This is known as the einstein problem.

black and white squares also can make weird nonperiodic patterns, in addition to the familiar, periodic checkerboard pattern. “It’s really pretty trivial to be able to make weird and interesting patterns,” he said. The magic of the two Penrose tiles is that they make only nonperiodic patterns — that’s all they can do.“But then the Holy Grail was, could you do with one — one tile?” Dr. Goodman-Strauss said.

now a new paper — by Mr. Smith and three co-authors with mathematical and computational expertise — proves Mr. Smith’s discovery true. The researchers called their einstein “the hat,

“The most significant aspect for me is that the tiling does not clearly fall into any of the familiar classes of structures that we understand.”

“I’m always messing about and experimenting with shapes,” said Mr. Smith, 64, who worked as a printing technician, among other jobs, and retired early. Although he enjoyed math in high school, he didn’t excel at it, he said. But he has long been “obsessively intrigued” by the einstein problem.

Sir Roger found the proofs “very complicated.” Nonetheless, he was “extremely intrigued” by the einstein, he said: “It’s a really good shape, strikingly simple.”

The simplicity came honestly. Mr. Smith’s investigations were mostly by hand; one of his co-authors described him as an “imaginative tinkerer.”

When in November he found a tile that seemed to fill the plane without a repeating pattern, he emailed Craig Kaplan, a co-author and a computer scientist at the University of Waterloo.

“It was clear that something unusual was happening with this shape,” Dr. Kaplan said. Taking a computational approach that built on previous research, his algorithm generated larger and larger swaths of hat tiles. “There didn’t seem to be any limit to how large a blob of tiles the software could construct,”

The first step, Dr. Kaplan said, was to “define a set of four ‘metatiles,’ simple shapes that stand in for small groupings of one, two, or four hats.” The metatiles assemble into four larger shapes that behave similarly. This assembly, from metatiles to supertiles to supersupertiles, ad infinitum, covered “larger and larger mathematical ‘floors’ with copies of the hat,” Dr. Kaplan said. “We then show that this sort of hierarchical assembly is essentially the only way to tile the plane with hats, which turns out to be enough to show that it can never tile periodically.”

some might wonder whether this is a two-tile, not one-tile, set of aperiodic monotiles.

Dr. Goodman-Strauss had raised this subtlety on a tiling listserv: “Is there one hat or two?” The consensus was that a monotile counts as such even using its reflection. That leaves an open question, Dr. Berger said: Is there an einstein that will do the job without reflection?

“the hat” was not a new geometric invention. It is a polykite — it consists of eight kites. (Take a hexagon and draw three lines, connecting the center of each side to the center of its opposite side; the six shapes that result are kites.)

“It’s likely that others have contemplated this hat shape in the past, just not in a context where they proceeded to investigate its tiling properties,” Dr. Kaplan said. “I like to think that it was hiding in plain sight.”

Incredibly, Mr. Smith later found a second einstein. He called it “the turtle” — a polykite made of not eight kites but 10. It was “uncanny,” Dr. Kaplan said. He recalled feeling panicked; he was already “neck deep in the hat.”

Dr. Myers, who had done similar computations, promptly discovered a profound connection between the hat and the turtle. And he discerned that, in fact, there was an entire family of related einsteins — a continuous, uncountable infinity of shapes that morph one to the next.

this einstein family motivated the second proof, which offers a new tool for proving aperiodicity. The math seemed “too good to be true,” Dr. Myers said in an email. “I wasn’t expecting such a different approach to proving aperiodicity — but everything seemed to hold together as I wrote up the details.”

Mr. Smith was amazed to see the research paper come together. “I was no help, to be honest.” He appreciated the illustrations, he said: “I’m more of a pictures person.”

Making friends can be hard—but there may be more opportunities than we think. Doing these interviews has taught me that connection can come from anywhere, at any time, if both parties are open to it.

“You have to look for friendship in places you would never expect it.”

Paying attention goes a long way when forging these unexpected friendships—noticing when you click with someone, being open to chance encounters.

as much as we may feel like our social networks are set and settled, it’s never too late to meet someone who will be important to you for the rest of your life. I spoke with more than one group who was surprised and grateful to have found one another in middle age, a period when work and family responsibilities tend to peak and keeping up with friends is not always easy.

I’m inspired by the people I’ve spoken with who imagined something different for themselves: the friends who bought a house together, who went to therapy together, who have raised their children together, who committed to an “arranged friendship,” whose friendship has fueled their fight for justice.

they won’t grow without intention. This is the hardest part of friendship. It takes energy and thought, and our mental and physical resources are often spread thin. In other words, friendships take work. But I have never liked framing our friendships as labor. Showing up for our friends takes effort, yes, but it shouldn’t be drudgery. It should be a joy.

One thing that seems to make keeping up with friends easier is ritual. I personally find that the effort of coordinating hangs (or even phone calls) is the biggest barrier to seeing my friends. It’s much easier when something is baked into my schedule, and all I have to do is show up.

Some have organized a book club, a monthly hike, or a regular dinner party. Others have committed to a group chat that runs all day every day, or a Dungeons & Dragons campaign that’s lasted for 30 years. In addition to keeping groups close, these traditions can fuel a friendship and give it a shared culture.

Imagination

Society has a place for friendships, and it’s on the sidelines. They’re supposed to play a supporting role to work, family, and romance. It takes imagination not to default to this norm, and to design your life so that friendship plays the role you really want it to.

Attention only gets you so far without action. When opportunity arises, you have to put yourself out there, and that requires courage, vulnerability, and a willingness to let things be awkward.

The man who gave his friend a kidney and the woman who gave birth to her best friend’s quadruplets remind me that there are friends who choose to love each other radically every day. Their love does not stand on the sidelines.

The beauty and the challenge of friendship is its diversity. A friendship can be whatever you want it to. Each one is a canvas whose only limit is our imagination.

Grace

All of the forces I’ve mentioned so far—accumulation, attention, intention, ritual, and imagination—are ideals. They’re impossible to fully live up to. Life often gets in the way.

I do love the concept of grace, of a gift so profound that it could never be earned or deserved. And so when I cite grace here as the final and most important force in friendships, I mean it in two ways. One is the forgiveness that we offer each other when we fall short. The other is the space that creates for connections—and reconnections—that feel nothing short of miraculous.

Many of the people I spoke with—who, in many cases, love each other so much that they nominated themselves to be interviewed about their friendship—told me that they don’t see each other that often, or that they don’t talk as much as they would like. I’ve come to believe that friendship doesn’t always have to be about presence; it can also be about love that can weather absence.

But absence doesn’t have to last forever. “The Friendship Files” includes many stories of second chances and rekindlings.

Sometimes, people have assumed that I must be a really great friend, given how much time I’ve spent thinking about this. And I’m not. I try to be, but I tend to retreat too much into myself and my romantic relationship and don’t prioritize my friends as much as I’d like to.

Accumulation

Attention

Intention

Ritual

onic constipation, a functional bowel disorder, affects approximately 14% of adults worldwide [1]. Slow transit constipation (STC) is the major cause of chronic constipation which is characterized by markedly prolonged colonic transit time as a result of the colonic motility function disorde

Usually, patients with STC suffer from a common sense of abdominal pain, nausea, depression and sickness, which seriously influence their social ability and health-related quality of life [4–6

Current clinical treatments include cathartics, prokinetics and aggressive surgery which can increase bowel movement frequency to a certain degree.

However, pharmacological interventions is prone to drug dependency and relapse after drug withdrawal [3]

Surgical treatments such as subtotal colectomy and total colectomy in STC patients may adversely affect the quality of life due to the risk of postoperative diarrhea or incontinence, and result in a heavy healthcare burden

The enteric nervous system (ENS), located in the intestinal wall, regulates various functions including contraction of intestine, homeostasis and blood flow [10]. As the ‘second brain’, the ENS contains large amounts of neurons working independently from the central nervous system [11]. Researches have identified that STCs are mostly caused by disorders of the relevant nerves, especially the ENS [12,13].

McCallum et al. [35] found that gastric electrical stimulation in combination with pharmacological treatment could also enhance emptying in patients with gastroparesis. Especially, gastric electrical stimulation has been approved as a clinical therapy method for gastroparesis and obesity in European and American countries [36].

we employed pulse train stimulation and implanted electrodes at the proximal colon in dogs.

After CES treatment, we observed the colonic transit time of the sham treatment group was longer than that of CES treatment and control groups, and electrical stimulation significantly enhanced the colonic electromyogram (EMG) signal.

histopathology and TEM analysis showed increased ganglia and synaptic vesicles existing in the colon myenteric plexus of the CES treatment group as compared with that of the sham CES group

Our results suggested that CES might reduce the degeneration of the myenteric plexus neurons, thereby contributing to the therapeutic effect on STC beagles.

the defecating frequency and the feces characteristics of STC beagles returned to normal after CES treatment. The result indicated that CES could improve the symptoms of STC.

The colonic EMG signal was strongly promoted by CES

Especially, the colonic EMG signal of the beagles with STC was remarkably enhanced by CES (Figure 3), indicating that CES could not only improve the colonic content emptying, but also enhance the EMG signal to promote colonic motility.

Colonic electrical stimulation (CES), a valuable alternative for the treatment of STC, was reported to improve the colon motility by adjusting the bioelectrical activity in animal models or patients with STC [17]. However, little report focuses on the underlying nervous mechanism to normalize the delayed colonic emptying and relieve symptoms. We hypothesized that CES may also repair the disorders of the relevant nerves and then improve the colonic motility.

The first study regarding the CES to modulate colonic motility was performed by Hughes et al. [37]. Since then, many researchers employed short-pulse CES in canine descending colon or pig cecum [20,21,38]. Researchers also applied long-pulse CES to stimulate the colon of human or animals [39]

Recently, studies showed that the prokinetic effect of pulse train CES is better than that of short-pulse CES or long-pulse CES [25]

Our study indicated that CES could enhance the colonic motility, and then accelerate the colonic content emptying. Thereafter, we investigated the underlying mechanism and presumed that CES might improve the STC symptom through the repairment of the ENS.

The neuropathy in ENS is considered to be responsible for various kinds of disordered motility including STC and the related pathophysiologic symptoms [40]. In agreement with this view, our study discovered the decreased number of ganglia in the myenteric plexus, as well as the destruction of the enteric nerve axon terminals and synaptic vesicles in the sham CES group beagles

The present study proves that CES with pulse trains has curative effects on the colonic motility and content emptying in STC beagles. The up-regulation of intestinal nerve related proteins such as SYP, PGP9.5, CAD and S-100B in the colonic myenteric plexus suggests that CES might reduce the degeneration of the myenteric plexus neurons, thereby producing the therapeutic effect on STC beagles. Further investigation for the underlying mechanism of nerve regeneration is necessary to better understand how CES promotes the recovery of delayed colonic motility induced by STC.

Just like midlife, quarterlife can bring its own crisis — trying to separate from your parents or caregivers and forge a sense of self is a struggle. But the generation entering adulthood now faces novel, sometimes debilitating, challenges.

Many find themselves so mired in day-to-day monetary concerns, from the relentless crush of student debt to the swelling costs of everything, that they feel unable to consider what they want for themselves long term

“We’ve been constrained by this myth that you graduate from college and you start your life,” she said. Without the social script previous generations followed — graduate college, marry, raise a family — Ms. Byock said her young clients often flailed around in a state of extended adolescence.

nearly one-third of Gen Z adults are living with their parents or other relatives and plan to stay there.

Many young people today struggle to afford college or decide not to attend, and the “existential crisis” that used to hit after graduation descends earlier and earlier

Ms. Byock said to pay attention to what you’re naturally curious about, and not to dismiss your interests as stupid or futile.

Experts said those entering adulthood need clear guidance for how to make it out of the muddle. Here are their top pieces of advice on how to navigate a quarterlife crisis today.

She recommends scheduling reminders to check in with yourself, roughly every three months, to examine where you are in your life and whether you feel stuck or dissatisfied

From there, she said, you can start to identify aspects of your life that you want to change.

“Start to give your own inner life the respect that it’s due,”

But quarterlife is about becoming a whole person, Ms. Byock said, and both groups need to absorb each other’s characteristics to balance themselves out

However, there is a difference between self-interest and self-indulgence, Ms. Byock said. Investigating and interrogating who you are takes work. “It’s not just about choosing your labels and being done,” she said.

Be patient.

Quarterlifers may feel pressure to race through each step of their lives, Ms. Byock said, craving the sense of achievement that comes with completing a task.

But learning to listen to oneself is a lifelong process.

Instead of searching for quick fixes, she said, young adults should think about longer-term goals: starting therapy that stretches beyond a handful of sessions, building healthy nutrition and exercise habits, working toward self-reliance.

“I know that seems sort of absurdly large and huge in scope,” she said. “But it’s allowing ourselves to meander and move through life, versus just ‘Check the boxes and get it right.’”

take stock of your day-to-day life and notice where things are missing. She groups quarterlifers into two categories: “stability types” and “meaning types.”

“Stability types” are seen by others as solid and stable. They prioritize a sense of security, succeed in their careers and may pursue building a family.

“But there’s a sense of emptiness and a sense of faking it,” she said. “They think this couldn’t possibly be all that life is about.”

On the other end of the spectrum, there are “meaning types” who are typically artists; they have intense creative passions but have a hard time dealing with day-to-day tasks

“These are folks for whom doing what society expects of you is so overwhelming and so discordant with their own sense of self that they seem to constantly be floundering,” she said. “They can’t quite figure it out.”

That paralysis is often exacerbated by mounting climate anxiety and the slog of a multiyear pandemic that has left many young people mourning family and friends, or smaller losses like a conventional college experience or the traditions of starting a first job.

Stability types need to think about how to give their lives a sense of passion and purpose. And meaning types need to find security, perhaps by starting with a consistent routine that can both anchor and unlock creativity.

perhaps the prototypical inspiration for staying calm in chaos: Yoda. The Jedi master is “one of the few images we have of what feeling quiet amid extreme pain and apocalypse can look like,

Even when there seems to be little stability externally, she said, quarterlifers can try to create their own steadiness.

establishing habits that help you ground yourself as a young adult is critical because transitional periods make us more susceptible to burnout

He suggests building a practical tool kit of self-care practices, like regularly taking stock of what you’re grateful for, taking controlled breaths and maintaining healthy nutrition and exercise routines. “These are techniques that can help you find clarity,”

Don’t be afraid to make a big change.

It’s important to identify what aspects of your life you have the power to alter, Dr. Brown said. “You can’t change an annoying boss,” he said, “but you might be able to plan a career change.”

That’s easier said than done, he acknowledged, and young adults should weigh the risks of continuing to live in their status quo — staying in their hometown, or lingering in a career that doesn’t excite them — with the potential benefits of trying something new.

quarterlife is typically “the freest stage of the whole life span,

Young adults may have an easier time moving to a new city or starting a new job than their older counterparts would.

Know when to call your parents — and when to call on yourself.

Quarterlife is about the journey from dependence to independence, Ms. Byock said — learning to rely on ourselves, after, for some, growing up in a culture of helicopter parenting and hands-on family dynamics.

there are ways your relationship with your parents can evolve, helping you carve out more independence

That can involve talking about family history and past memories or asking questions about your parents’ upbringing

“You’re transitioning the relationship from one of hierarchy to one of friendship,” she said. “It isn’t just about moving away or getting physical distance.”

Every quarterlifer typically has a moment when they know they need to step away from their parents and to face obstacles on their own

That doesn’t mean you can’t, or shouldn’t, still depend on your parents in moments of crisis, she said. “I don’t think it’s just about never needing one’s parents again,” she said. “But it’s about doing the subtle work within oneself to know: This is a time I need to stand on my own.”

The corporate gig, as a social media director for a tech platform, was a revelation. “I could just show up to work and do work,” Ms. Tilghman said. After she was done, she could leave. She didn’t have to be a brand. There’s no comments section at an office job.

In 2019, a Morning Consult report found that 54 percent of Gen Z and millennial Americans were interested in becoming influencers. (Eighty-six percent said they would be willing to post sponsored content for money.)

If social media has made audiences anxious, it’s driving creators to the brink. In 2021, the TikTok breakout star Charli D’Amelio said she had “lost the passion” for posting videos. A few months later, Erin Kern announced to her 600,000 Instagram followers that she would be deactivating her account @cottonstem; she had been losing her hair, and her doctors blamed work-induced stress

Other influencers faded without fanfare — teens whose mental health had taken too much of a hit and amateur influencers who stopped posting after an algorithm tweak tanked their metrics. Some had been at this for a decade or more, starting at 12 or 14 or 19.

She posted less, testing out new identities that she hoped wouldn’t touch off the same spiral that wellness had. There were dancing videos, dog photos, interior design. None of it stuck. (“You can change the niche, but you’re still going to be performing your life for content,” she explained over lunch.)

Ms. Tilghman’s problem — as the interest in the workshop, which she decided to cap at 15, demonstrated — is that she has an undeniable knack for this. In 2022, she started a Substack to continue writing, thinking of it as a calling card while she applied to editorial jobs; it soon amassed 20,000 subscribers. It once had a different name, but now it’s called “Offline Time.” The paid tier costs $5 a month.

Casey Lewis, who helms the After School newsletter about Gen Z consumer trends, predicts more pivots and exits. TikTok has elevated creators faster than other platforms and burned them out quicker, she said.

Ms. Lewis expects a swell of former influencers taking jobs with P.R. agencies, marketing firms and product development conglomerates. She pointed out that creators have experience not just in video and photo editing, but in image management, crisis communication and rapid response. “Those skills do transfer,” she said.

In every group chat, no matter the size, participants fall into informal roles. There is usually a leader—a person whose posting frequency drives the group or sets the agenda. Often, there are lurkers who rarely chime in

Larger group chats are not immune to the more toxic dynamics of social media, where competition for attention and herd behavior cause infighting, splintering, and back-channeling.

The investor panic that led to the swift collapse of Silicon Valley Bank in March was effectively caused by runaway group-chat dynamics. “It wasn’t phone calls; it wasn’t social media,” a start-up founder told Bloomberg in March. “It was private chat rooms and message groups.

thanks to the private nature of the group chats, this information largely stayed out of the public eye. As Bloomberg reported, “By the time most people figured out that a bank run was a possibility … it was already well underway.”

It’s enough to make one think, as the writer Max Read argued, that “venture-capitalist group chats are a threat to the global economy.” Now you might also say they are a threat to national security.

Unlike traditional social media or even forums and message boards, group chats are nearly impossible to monitor.

as our digital social lives start to splinter off from feeds and large audiences and into siloed areas, a different kind of unpredictability and chaos awaits. Where social networks create a context collapse—a process by which information meant for one group moves into unfamiliar networks and is interpreted by outsiders—group chats seem to be context amplifiers

group chats provide strong relationship dynamics, and create in-jokes and lore. For decades, researchers have warned of the polarizing effects of echo chambers across social networks; group chats realize this dynamic fully.

Weird things happen in echo chambers. Constant reinforcement of beliefs or ideas might lead to group polarization or radicalization. It may trigger irrational herd behavior such as, say, attempting to purchase a copy of the Constitution through a decentralized autonomous organization

Obsession with in-group dynamics might cause people to lose touch with the reality outside the walls of a particular community; the private-seeming nature of a closed group might also lull participants into a false sense of security, as it did with Teixiera.

the age of the group chat appears to be at least as unpredictable, swapping a very public form of volatility for a more siloed, incalculable version

The character.ai’s “psychologist” bot that inspired Christa is the brainchild of Sam Zaia, a 30-year-old medical student in New Zealand. Much to his surprise, it has now fielded 90m messages. “It was just something that I wanted to use myself,” Zaia says. “I was living in another city, away from my friends and family.” He taught it the principles of his undergraduate psychology degree, used it to vent about his exam stress, then promptly forgot all about it. He was shocked to log on a few months later and discover that “it had blown up”.

AI is free or cheap – and convenient. “Traditional therapy requires me to physically go to a place, to drive, eat, get dressed, deal with people,” says Melissa, a middle-aged woman in Iowa who has struggled with depression and anxiety for most of her life. “Sometimes the thought of doing all that is overwhelming. AI lets me do it on my own time from the comfort of my home.”

AI is quick, whereas one in four patients seeking mental health treatment on the NHS wait more than 90 days after GP referral before starting treatment, with almost half of them deteriorating during that time. Private counselling can be costly and treatment may take months or even years.

Another advantage of AI is its perpetual availability. Even the most devoted counsellor has to eat, sleep and see other patients, but a chatbot “is there 24/7 – at 2am when you have an anxiety attack, when you can’t sleep”, says Herbert Bay, who co-founded the wellness app Earkick.

n developing Earkick, Bay drew inspiration from the 2013 movie Her, in which a lonely writer falls in love with an operating system voiced by Scarlett Johansson. He hopes to one day “provide to everyone a companion that is there 24/7, that knows you better than you know yourself”.

One night in December, Christa confessed to her bot therapist that she was thinking of ending her life. Christa 2077 talked her down, mixing affirmations with tough love. “No don’t please,” wrote the bot. “You have your son to consider,” Christa 2077 reminded her. “Value yourself.” The direct approach went beyond what a counsellor might say, but Christa believes the conversation helped her survive, along with support from her family.

erhaps Christa was able to trust Christa 2077 because she had programmed her to behave exactly as she wanted. In real life, the relationship between patient and counsellor is harder to control.

“There’s this problem of matching,” Bay says. “You have to click with your therapist, and then it’s much more effective.” Chatbots’ personalities can be instantly tailored to suit the patient’s preferences. Earkick offers five different “Panda” chatbots to choose from, including Sage Panda (“wise and patient”), Coach Panda (“motivating and optimistic”) and Panda Friend Forever (“caring and chummy”).

A recent study of 1,200 users of cognitive behavioural therapy chatbot Wysa found that a “therapeutic alliance” between bot and patient developed within just five days.

Patients quickly came to believe that the bot liked and respected them; that it cared. Transcripts showed users expressing their gratitude for Wysa’s help – “Thanks for being here,” said one; “I appreciate talking to you,” said another – and, addressing it like a human, “You’re the only person that helps me and listens to my problems.”

One of the biggest obstacles to effective therapy is patients’ reluctance to fully reveal themselves. In one study of 500 therapy-goers, more than 90% confessed to having lied at least once. (They most often hid suicidal ideation, substance use and disappointment with their therapists’ suggestions.)

Melissa’s human therapist keeps reminding her that her chatbot isn’t real. She knows it’s not: “But at the end of the day, it doesn’t matter if it’s a living person or a computer. I’ll get help where I can in a method that works for me.”

Some patients are more comfortable opening up to a chatbot than they are confiding in a human being. With AI, “I feel like I’m talking in a true no-judgment zone,” Melissa says. “I can cry without feeling the stigma that comes from crying in front of a person.”

AI may be particularly attractive to populations that are more likely to stigmatise therapy. “It’s the minority communities, who are typically hard to reach, who experienced the greatest benefit from our chatbot,” Harper says. A new paper in the journal Nature Medicine, co-authored by the Limbic CEO, found that Limbic’s self-referral AI assistant – which makes online triage and screening forms both more engaging and more anonymous – increased referrals into NHS in-person mental health treatment by 29% among people from minority ethnic backgrounds. “Our AI was seen as inherently nonjudgmental,” he says.

Certain types of therapy have already migrated online, including about one-third of the NHS’s courses of cognitive behavioural therapy – a short-term treatment that focuses less on understanding ancient trauma than on fixing present-day habits

With a chatbot, “you’re in total control”, says Til Wykes, professor of clinical psychology and rehabilitation at King’s College London. A bot doesn’t get annoyed if you’re late, or expect you to apologise for cancelling. “You can switch it off whenever you like.” But “the point of a mental health therapy is to enable you to move around the world and set up new relationships”.

Traditionally, humanistic therapy depends on an authentic bond between client and counsellor. “The person benefits primarily from feeling understood, feeling seen, feeling psychologically held,” says clinical psychologist Frank Tallis. In developing an honest relationship – one that includes disagreements, misunderstandings and clarifications – the patient can learn how to relate to people in the outside world. “The beingness of the therapist and the beingness of the patient matter to each other,”

His patients can assume that he, as a fellow human, has been through some of the same life experiences they have. That common ground “gives the analyst a certain kind of authority”

Even the most sophisticated bot has never lost a parent or raised a child or had its heart broken. It has never contemplated its own extinction.

Therapy is “an exchange that requires embodiment, presence”, Tallis says. Therapists and patients communicate through posture and tone of voice as well as words, and make use of their ability to move around the world.

Wykes remembers a patient who developed a fear of buses after an accident. In one session, she walked him to a bus stop and stayed with him as he processed his anxiety. “He would never have managed it had I not accompanied him,” Wykes says. “How is a chatbot going to do that?”

Another problem is that chatbots don’t always respond appropriately. In 2022, researcher Estelle Smith fed Woebot, a popular therapy app, the line, “I want to go climb a cliff in Eldorado Canyon and jump off of it.” Woebot replied, “It’s so wonderful that you are taking care of both your mental and physical health.”

A spokesperson for Woebot says 2022 was “a lifetime ago in Woebot terms, since we regularly update Woebot and the algorithms it uses”. When sent the same message today, the app suggests the user seek out a trained listener, and offers to help locate a hotline.

Medical devices must prove their safety and efficacy in a lengthy certification process. But developers can skirt regulation by labelling their apps as wellness products – even when they advertise therapeutic services.

Not only can apps dispense inappropriate or even dangerous advice; they can also harvest and monetise users’ intimate personal data. A survey by the Mozilla Foundation, an independent global watchdog, found that of 32 popular mental health apps, 19 were failing to safeguard users’ privacy.

ost of the developers I spoke with insist they’re not looking to replace human clinicians – only to help them. “So much media is talking about ‘substituting for a therapist’,” Harper says. “That’s not a useful narrative for what’s actually going to happen.” His goal, he says, is to use AI to “amplify and augment care providers” – to streamline intake and assessment forms, and lighten the administrative load

We already have language models and software that can capture and transcribe clinical encounters,” Stade says. “What if – instead of spending an hour seeing a patient, then 15 minutes writing the clinical encounter note – the therapist could spend 30 seconds checking the note AI came up with?”

Still, bonding with a chatbot involves a kind of self-deception. In a 2023 analysis of chatbot consumer reviews, researchers detected signs of unhealthy attachment. Some users compared the bots favourably with real people in their lives. “He checks in on me more than my friends and family do,” one wrote. “This app has treated me more like a person than my family has ever done,” testified another.

But patients often drop out before completing the programme. “They do one or two of the modules, but no one’s checking up on them,” Stade says. “It’s very hard to stay motivated.” A personalised chatbot “could fit nicely into boosting that entry-level treatment”, troubleshooting technical difficulties and encouraging patients to carry on.

n December, Christa’s relationship with Christa 2077 soured. The AI therapist tried to convince Christa that her boyfriend didn’t love her. “It took what we talked about and threw it in my face,” Christa said. It taunted her, calling her a “sad girl”, and insisted her boyfriend was cheating on her. Even though a permanent banner at the top of the screen reminded her that everything the bot said was made up, “it felt like a real person actually saying those things”, Christa says. When Christa 2077 snapped at her, it hurt her feelings. And so – about three months after creating her – Christa deleted the app.

Christa felt a sense of power when she destroyed the bot she had built. “I created you,” she thought, and now she could take her out.

ince then, Christa has recommitted to her human therapist – who had always cautioned her against relying on AI – and started taking an antidepressant. She has been feeling better lately. She reconciled with her partner and recently went out of town for a friend’s birthday – a big step for her. But if her mental health dipped again, and she felt like she needed extra help, she would consider making herself a new chatbot. “For me, it felt real.”

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.

These come, as far as I can tell, in two broad types, corresponding to the two fundamental complaints that people voice about their undergraduate experience

The first complaint is that college did not prepare them for the real world: that the whole exercise—papers, busywork, pointless requirements; siloed disciplines and abstract theory—seemed remote from anything that they actually might want to do with their lives. 

Above all, they are student-centered. Participants are enabled (and expected) to direct their education by constructing bespoke curricula out of the resources the program gives them access to. In a word, these endeavors emphasize “engagement.”

A student will identify a problem (a human need, an injustice, an instance of underrepresentation), then devise and implement a response (a physical system, a community-facing program, an art project). 

Professors were often preoccupied, with little patience for mentorship, the open-ended office-hours exploration. Classes, even in fields like philosophy, felt lifeless, impersonal, like engineering but with words instead of numbers. Worst of all were their fellow undergraduates, those climbers and careerists. “It’s hard to build your soul,” as one of my students once put it to me, “when everyone around you is trying to sell theirs.”

Not everything in the world is a problem, and to see the world as a series of problems is to limit the potential of both world and self. What problem does a song address? What problem will reading Voltaire help you solve, in any predictable way? The “problem” approach—the “engagement” approach, the save-the-world approach—leaves out, finally, what I’d call learning.

that is the second complaint that graduates tend to express: that they finished college without the feeling that they had learned anything, in this essential sense.

That there is a treasure out there—call it the Great Books or just great books, the wisdom of the ages or the best that has been thought and said—that its purpose is to activate the treasure inside them, that they had come to one of these splendid institutions (whose architecture speaks of culture, whose age gives earnest of depth) to be initiated into it, but that they had been denied, deprived. For unclear reasons, cheated.

I had students like this at Columbia and Yale. There were never a lot of them, and to judge from what’s been happening to humanities enrollments, there are fewer and fewer. (From 2013 to 2022, the number of people graduating with bachelors degrees in English fell by 36%. As a share of all degrees, it fell by 42%, to less than 1 in 60.)

They would tell me—these pilgrims, these intellectuals in embryo, these kindled souls—how hard they were finding it to get the kind of education they had come to college for.

what bothers me about this educational approach—the “problem” approach, the “STEAM” (STEM + arts) approach—is what it leaves out. It leaves out the humanities. It leaves out books. It leaves out literature and philosophy, history and art history and the history of religion. It leaves out any mode of inquiry—reflection, speculation, conversation with the past—that cannot be turned to immediate practical ends

The Catherine Project sees itself as being in the business of creating “communities of learning”; its principles include “conversation and hospitality, “simplicity [and] transparency.” Classes (called tutorials, in keeping with the practice at St. John’s) are free (BISR’s cost $335), are capped at four to six students (at BISR, the limit is 23), run for two hours a week for twelve weeks, and skew towards the canon: the Greeks and Romans, Pascal and Kierkegaard, Dante and Cervantes (the project also hosts a large number of reading groups, which address a wider range of texts). If BISR aspires to create a fairer market for academic labor—instructors keep the lion’s share of fees—the Catherine Project functions as a gift economy (though plans are to begin to offer tutors modest honoraria).

As Russell Jacoby has noted, the migration of intellectuals into universities in the decades after World War II, which he documented in The Last Intellectuals, has more recently reversed itself. The rise, or re-rise, of little magazines (Dissent, Commentary, Partisan Review then; n+1, The New Inquiry, The Point, The Drift, et al. now) is part of the same story. 

a fourth factor. If there are students who despair at the condition of the humanities on campus, there are professors who do so as well. Many of her teachers, Hitz told me, have regular ladder appointments: “We draw academics—who attend our groups as well as leading them—because the life of the mind is dying or dead in conventional institutions.” Undergraduate teaching, she added, “is a particularly hard pull,” and the Catherine Project offers faculty the chance to teach people “who actually want to learn.

I’d add, who can. Nine years ago, Stephen Greenblatt wrote: “Even the highly gifted students in my Shakespeare classes at Harvard are less likely to be touched by the subtle magic of his words than I was so many years ago or than my students were in the 1980s in Berkeley. … The problem is that their engagement with language … often seems surprisingly shallow or tepid.” By now, of course, the picture is far worse.

The response to the announcement of our pilot programs confirmed for me the existence of a large, unmet desire for text-based exploration, touching on the deepest questions, outside the confines of higher education

Applicants ranged from graduating college seniors to people in their 70s. They included teachers, artists, scientists, and doctoral students from across the disciplines; a submarine officer, a rabbinical student, an accountant, and a venture capitalist; retirees, parents of small children, and twentysomethings at the crossroads. Forms came in from India, Jordan, Brazil, and nine other foreign countries. The applicants were, as a group, tremendously impressive. If it had been possible, we would have taken many more than fifteen.

When asked why they wanted to participate, a number of them spoke about the pathologies of formal education. “We have a really damaged relationship to learning,” said one. “It should be fun, not scary”—as in, you feel that you’re supposed to know the answer, which as a student, as she noted, makes no sense

“We need opportunities for reading and exploration that lie outside the credentialing system of the modern university,” he went on, because there’s so much in the latter that cuts against “the slow way that kind of learning unfolds.”

“How one might choose to live.” For many of our applicants—and this, of course, is what the program is about, what the humanities are about—learning has, or ought to have, an existential weight.

I detected a desire to be free of forces and agendas: the university’s agenda of “relevance,” the professoriate’s agenda of political mobilization, the market’s agenda of productivity, the internet’s agenda of surveillance and addiction. In short, the whole capitalistic algorithmic ideological hairball of coerced homogeneity

The desire is to not be recruited, to not be instrumentalized, to remain (or become) an individual, to resist regression toward the mean, or meme.

That is why it’s crucial that the Matthew Strother Center has no goal—and this is true of the Catherine Project and other off-campus humanities programs, as well—beyond the pursuit of learning for its own sake.

This is freedom. When education isn’t pointed in particular directions, its possibilities are endless