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Understanding how empathy can immobilize us like that is a critical step for helping others — and ourselves.

Early researchers labeled it compassion fatigue and described it as the cost of caring.

Having concluded that nothing they do will make a difference, they start to become indifferent.

The symptoms of empathic distress were originally diagnosed in health care, with nurses and doctors who appeared to become insensitive to the pain of their patients.

Empathic distress explains why many people have checked out in the wake of these tragedies

when two neuroscientists, Olga Klimecki and Tania Singer, reviewed the evidence, they discovered that “compassion fatigue” is a misnomer. Caring itself is not costly. What drains people is not merely witnessing others’ pain but feeling incapable of alleviating it.

In times of sustained anguish, empathy is a recipe for more distress, and in some cases even depression. What we need instead is compassion.

empathy and compassion aren’t the same. Empathy absorbs others’ emotions as your own: “I’m hurting for you.”

Compassion focuses your action on their emotions: “I see that you’re hurting, and I’m here for you.”

“Empathy is biased,” the psychologist Paul Bloom writes. It’s something we usually reserve for our own group, and in that sense, it can even be “a powerful force for war and atrocity.”

Dr. Singer and their colleagues trained people to empathize by trying to feel other people’s pain. When the participants saw someone suffering, it activated a neural network that would light up if they themselves were in pain. It hurt. And when people can’t help, they escape the pain by withdrawing.

To combat this, the Klimecki and Singer team taught their participants to respond with compassion rather than empathy — focusing not on sharing others’ pain but on noticing their feelings and offering comfort.

A different neural network lit up, one associated with affiliation and social connection. This is why a growing body of evidence suggests that compassion is healthier for you and kinder to others than empathy:

When you see others in pain, instead of causing you to get overloaded and retreat, compassion motivates you to reach out and help

The most basic form of compassion is not assuaging distress but acknowledging it.

in my research, I’ve found that being helpful has a secondary benefit: It’s an antidote to feeling helpless.

To figure out who needs your support after something terrible happens, the psychologist Susan Silk suggests picturing a dart board, with the people closest to the trauma in the bull’s-eye and those more peripherally affected in the outer rings.

Once you’ve figured out where you belong on the dart board, look for support from people outside your ring, and offer it to people closer to the center.

Even if people aren’t personally in the line of fire, attacks targeting members of a specific group can shatter a whole population’s sense of security.

If you notice that people in your life seem disengaged around an issue that matters to you, it’s worth considering whose pain they might be carrying.

Instead of demanding that they do more, it may be time to show them compassion — and help them find compassion for themselves, too.

Your small gesture of kindness won’t end the crisis in the Middle East, but it can help someone else. And that can give you the strength to help more.

People believe that memory should be effortless, but their expectations for how much they should remember are totally out of whack with how much they’re capable of remembering.1

What is the most common misconception about memory?

Another misconception is that memory is supposed to be an archive of the past. We expect that we should be able to replay the past like a movie in our heads.

we don’t replay the past as it happened; we do it through a lens of interpretation and imagination.

How much are we capable of remembering, from both an episodic2 2 Episodic memory is the term for the memory of life experiences. and a semantic3 3 Semantic memory is the term for the memory of facts and knowledge about the world. standpoint?

I would argue that we’re all everyday-memory experts, because we have this exceptional semantic memory, which is the scaffold for episodic memory.

If what we’re remembering, or the emotional tenor of what we’re remembering, is dictated by how we’re thinking in a present moment, what can we really say about the truth of a memory?

But if memories are malleable, what are the implications for how we understand our “true” selves?

your question gets to a major purpose of memory, which is to give us an illusion of stability in a world that is always changing. Because if we look for memories, we’ll reshape them into our beliefs of what’s happening right now. We’ll be biased in terms of how we sample the past. We have these illusions of stability, but we are always changing

And depending on what memories we draw upon, those life narratives can change.

we have this illusion that much of the world is cause and effect. But the reason, in my opinion, that we have that illusion is that our brain is constantly trying to find the patterns

One thing that makes the human brain so sophisticated is that we have a longer timeline in which we can integrate information than many other species. That gives us the ability to say: “Hey, I’m walking up and giving money to the cashier at the cafe. The barista is going to hand me a cup of coffee in about a minute or two.”

There is this illusion that we know exactly what’s going to happen, but the fact is we don’t. Memory can overdo it: Somebody lied to us once, so they are a liar; somebody shoplifted once, they are a thief.

If people have a vivid memory of something that sticks out, that will overshadow all their knowledge about the way things work. So there’s kind of an illus

I know it sounds squirmy to say, “Well, I can’t answer the question of how much we remember,” but I don’t want readers to walk away thinking memory is all made up.

I think of memory more like a painting than a photograph. There’s often photorealistic aspects of a painting, but there’s also interpretation. As a painter evolves, they could revisit the same subject over and over and paint differently based on who they are now. We’re capable of remembering things in extraordinary detail, but we infuse meaning into what we remember. We’re designed to extract meaning from the past, and that meaning should have truth in it. But it also has knowledge and imagination and, sometimes, wisdom.

memory, often, is educated guesses by the brain about what’s important. So what’s important? Things that are scary, things that get your desire going, things that are surprising. Maybe you were attracted to this person, and your eyes dilated, your pulse went up. Maybe you were working on something in this high state of excitement, and your dopamine was up.

It could be any of those things, but they’re all important in some way, because if you’re a brain, you want to take what’s surprising, you want to take what’s motivationally important for survival, what’s new.

On the more intentional side, are there things that we might be able to do in the moment to make events last in our memories? In some sense, it’s about being mindful. If we want to form a new memory, focus on aspects of the experience you want to take with you.

If you’re with your kid, you’re at a park, focus on the parts of it that are great, not the parts that are kind of annoying. Then you want to focus on the sights, the sounds, the smells, because those will give you rich detail later on

Another part of it, too, is that we kill ourselves by inducing distractions in our world. We have alerts on our phones. We check email habitually.

When we go on trips, I take candid shots. These are the things that bring you back to moments. If you capture the feelings and the sights and the sounds that bring you to the moment, as opposed to the facts of what happened, that is a huge part of getting the best of memory.

this goes back to the question of whether the factual truth of a memory matters to how we interpret it. I think it matters to have some truth, but then again, many of the truths we cling to depend on our own perspective.

There’s a great experiment on this. These researchers had people read this story about a house.8 8 The study was “Recall of Previously Unrecallable Information Following a Shift in Perspective,” by Richard C. Anderson and James W. Pichert. One group of subjects is told, I want you to read this story from the perspective of a prospective home buyer. When they remember it, they remember all the features of the house that are described in the thing. Another group is told, I want you to remember this from the perspective of a burglar. Those people tend to remember the valuables in the house and things that you would want to take. But what was interesting was then they switched the groups around. All of a sudden, people could pull up a number of details that they didn’t pull up before. It was always there, but they just didn’t approach it from that mind-set. So we do have a lot of information that we can get if we change our perspective, and this ability to change our perspective is exceptionally important for being accurate. It’s exceptionally important for being able to grow and modify our beliefs

Some of the worst leadership decisions of our time can be traced to escalation of commitment. Many people lost their lives because American presidents pursued a futile war in Vietnam — and continued searching for weapons of mass destruction that weren’t in Iraq.

Escalation of commitment helps to explain why leaders are often so reluctant to loosen their grip on power. Losing a high-status position can make them feel as if they’re losing their place in the world. It leaves them with bruised egos and wounded pride.

we use our big brains not to make rational decisions, but rather to rationalize the decisions we’ve already made

Escalation is likely when people are directly responsible for and publicly attached to a decision, when it has been a long journey and the end is in sight, and when they have reasons to be confident that they can succeed.

President Biden’s current situation checks all those boxes

the people closest to a leader are precisely the ones who are most susceptible to confirmation bias. They’re too personally invested in his success and too likely to dismiss warning signs.

What Mr. Biden needs is not a support network but a challenge network — people who have the will to put the country’s interests ahead of his and the skill to coldly assess his chances.

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.