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The study analyzed friendship ties among 84 subjects (ages 23 to 38) in a business management class by asking them to rank one another on a five-point continuum of closeness from “I don’t know this person” to “One of my best friends.” The feelings were mutual 53 percent of the time while the expectation of reciprocity was pegged at 94 percent. This is consistent with data from several other friendship studies conducted over the past decade, encompassing more than 92,000 subjects, in which the reciprocity rates ranged from 34 percent to 53 percent.

“Friendship is difficult to describe,” said Alexander Nehamas, a professor of philosophy at Princeton, who in his latest book, “On Friendship,” spends almost 300 pages trying to do just that. “It’s easier to say what friendship is not and, foremost, it is not instrumental.”

It is not a means to obtain higher status, wangle an invitation to someone’s vacation home or simply escape your own boredom. Rather, Mr. Nehamas said, friendship is more like beauty or art, which kindles something deep within us and is “appreciated for its own sake.

“Treating friends like investments or commodities is anathema to the whole idea of friendship,” said Ronald Sharp, a professor of English at Vassar College, who teaches a course on the literature of friendship. “It’s not about what someone can do for you, it’s who and what the two of you become in each other’s presence.”

Some blame human beings’ basic optimism, if not egocentrism, for the disconnect between perceived and actual friendships. Others point to a misunderstanding of the very notion of friendship in an age when “friend” is used as a verb, and social inclusion and exclusion are as easy as a swipe or a tap on a smartphone screen.

By his definition, friends are people you take the time to understand and allow to understand you.

Because time is limited, so, too, is the number of friends you can have, according to the work of the British evolutionary psychologist Robin I.M. Dunbar. He describes layers of friendship, where the topmost layer consists of only one or two people, say a spouse and best friend with whom you are most intimate and interact daily. The next layer can accommodate at most four people for whom you have great affinity, affection and concern and who require weekly attention to maintain. Out from there, the tiers contain more casual friends with whom you invest less time and tend to have a less profound and more tenuous connection. Without consistent contact, they easily fall into the realm of acquaintance. You may be friendly with them but they aren’t friends.

“There is a limited amount of time and emotional capital we can distribute, so we only have five slots for the most intense type of relationship,” Mr. Dunbar said. “People may say they have more than five but you can be pretty sure they are not high-quality friendships.

Such boasting implies they have soul mates to spare in a culture where we are taught that leaning on someone is a sign of weakness and power is not letting others affect you. But friendship requires the vulnerability of caring as well as revealing things about yourself that don’t match the polished image in your Facebook profile or Instagram feed, said Mr. Nehamas at Princeton. Trusting that your bond will continue, and might even be strengthened, despite your shortcomings and inevitable misfortunes, he said, is a risk many aren’t willing to take.

According to medical experts, playing it safe by engaging in shallow, unfulfilling or nonreciprocal relationships has physical repercussions. Not only do the resulting feelings of loneliness and isolation increase the risk of death as much as smoking, alcoholism and obesity; you may also lose tone, or function, in the so-called smart vagus nerve, which brain researchers think allows us to be in intimate, supportive and reciprocal relationships in the first place.

In the presence of a true friend, Dr. Banks said, the smart or modulating aspect of the vagus nerve is what makes us feel at ease rather than on guard as when we are with a stranger or someone judgmental. It’s what enables us to feel O.K. about exposing the soft underbelly of our psyche and helps us stay engaged and present in times of conflict. Lacking authentic friendships, the smart vagus nerve is not exercised. It loses tone and one’s anxiety remains high, making abiding, deep connections difficult.

Worse still, the laws of physics themselves seem to be working against us. Ours isn’t just a randomly hostile universe, it's an actively hostile universe

The history of physics, in fact, is a marvel of using simple symmetry principles to construct complicated laws of the universe

if the entire universe were made symmetric, then all of the good features (e.g., you) are decidedly asymmetric lumps that ruin the otherwise perfect beauty of the cosmo

it would be a mistake to be comforted by the symmetries of the universe. In truth, they are your worst enemies. Everything we know about those rational, predictable arrangements dictates that you shouldn't be here at all.

Matter and antimatter should have completely annihilated one another in the first nanoseconds after the Big Bang. You should not even exist. But you do, and there’s lots more matter where you came from.

The symmetry of the universe would bake us in no time at all, but an asymmetry rescues us

In literally every experiment and observation that we’ve ever done, matter and antimatter get created (or annihilated) in perfect concert. That is, every experiment except for one: us.

How hostile is the universe to your fundamental existence? Very. Even the simplest assumptions about our place in the universe seem to lead inexorably to devastating results

if the perfect symmetry between matter and antimatter remained perfect, you wouldn’t be here to think about it.

The flow of time (as near as we can tell) is completely arbitrary. Does entropy increase with time or does it make time? Are our memories the thing that ultimately breaks the symmetry of time?

It seems only a matter of luck (and some fairly arbitrary-looking math) that a symmetric universe would end up being remotely hospitable to complex creatures like us

Without electrons binding to protons, there would be no chemistry, no molecules, and nothing more complicated than a cloud of charged gas. And you’re not a sentient cloud of gas, are you?

While science and philosophy do at times overlap, they are fundamentally different approaches to understanding. So philosophers should not add to the conceptual confusion that subsumes all knowledge into science.

various disciplines we ordinarily treat as science are at least as — if not more —philosophical than scientific. Take for example mathematics, theoretical physics, psychology and economics. These are predominately rational conceptual disciplines. That is, they are not chiefly reliant on empirical observation. For unlike science, they may be conducted while sitting in an armchair with eyes closed.

unlike empirical observations, which may be mistaken or incomplete, philosophical findings depend primarily on rational and logical principles. As such, whereas science tends to alter and update its findings day to day through trial and error, logical deductions are timeless.

while mathematics is empirically testable at such rudimentary levels, it stops being so in its purest forms, like analysis and number theory. Proofs in these areas are conducted entirely conceptually

Logically fallacious arguments can be rather sophisticated and persuasive. But they are nevertheless invalid and always will be. Exposing such errors is part of philosophy’s stock and trade.

in ethics, science cannot necessarily tell us what to value

Ultimately as a result of Wittgenstein’s philosophy, we know that natural language is a public phenomenon that cannot logically be invented in isolation.

These are essentially conceptual clarifications. And as such, they are relatively timeless philosophical truths.

This is also why jurisprudence qualifies as an objective body of knowledge

Supreme Court justices are not so much scientific as philosophical experts on the nature of justice. And that is not to say their expertise does not count as genuine knowledge. In the best cases, it rises to the loftier level of wisdom

Though philosophy does sometimes employ thought experiments, these aren’t actually scientific, for they are conducted entirely in the imagination.

Wittgenstein showed that an ordinary word such as “game” is used consistently in myriad contrasting ways without possessing any essential unifying definition. Though this may seem impossible, the meaning of such terms is actually determined by their contextual usage

evidence of how most people happen to be does not necessarily tell us everything about how we should aspire to be. For how we should aspire to be is a conceptual question, namely, of how we ought to act, as opposed to an empirical question of how we do act.

if you can think of times in your life that you've treated people with extraordinary decency and love, and pure uninterested concern, just because they were valuable as human beings. The ability to do that with ourselves. To treat ourselves the way we would treat a really good, precious friend. Or a tiny child of ours that we absolutely loved more than life itself. And I think it's probably possible to achieve that. I think part of the job we're here for is to learn how to do it.

The term for what he's describing, in Christian theology, is grace.

Formal schooling is terribly important; it helps you think in the way that IQ testers like.

In 1910, schools were focused on kids memorizing things about the real world. Today, they are entirely about relationships.

One of the fundamental things is the switch from “utilitarian spectacles” to “scientific spectacles.” The fact that we wear scientific spectacles doesn’t mean that we actually know a lot about science.

in 1900 in America, if you asked a child, what do dogs and rabbits have in common, they would say, “Well, you use dogs to hunt rabbits.” This is not the answer that the IQ tests want. They want you to classify. Today, a child would be likely to say, “They are both animals.” They picked up the habit of classification and use the vocabulary of science.

we have learned to use logic to attack the hypothetical. We have an ability to deal with a much wider range of problems than our ancestors would.

In 1950, teenagers could not only understand their parents, but they could also mimic their speech. Today, teenagers can still understand their parents. Their passive vocabularies are good enough. But when it comes to the words they actively use, they are much less capable of adult speak.

The brighter you are, the quicker after the age of 65 you have a downward curve for your analytic abilities

Retire from your job, but read great literature. Read about the history of science. Try and keep up your problem solving skills

One of the most interesting predictions is what will happen to the developing world. If they industrialize, in theory, they should have the explosive IQ gains in the coming century that we had in the last century.

1. Art is not an abstract category.

You do not stand in front of "Art" in a museum or a gallery, you stand in front of particular painting. The artist Barnett Newman once quipped, "aesthetics is for artists what ornithology is for birds."

2. Art is not a political weapon

The value of art is not found in its capacity to effect political change "out there" but to work on us—you and me—to connect with our pain and suffering, hope and yearning, to have the run of our "inner chambers,"

Any cultural practice that is worth taking seriously, that has a history and tradition, whether cooking, ballroom dancing, or chess, takes practice, and thus requires effort to learn

3. Art is not easy

Each work of art is a response to a conversation that spans centuries, each artist receives and passes on this tradition in their own distinctive way

4. Art is not a visual illustration of the artist’s worldview

No human being possesses a unified "worldview" that is manifest in and through each of her intentional acts or artifacts she produces

An artist does not paint a picture to express what she already knows or believes. She paints to learn something about herself and the world—something she doesn’t already know.

5. Art does not form virtue

Art always pushes against the pragmatism, moralism, and utilitarianism that shapes life inside as well as outside the church. It starts with our weakness, desperation, and brokenness in its search for hope and beauty and stakes its existence and relevance on the belief that all appearances deceive.

The author retorts “I really have an issue with the word hoax.” He regards himself as a performance artist. His response – “It’s the people who reported it who are deceiving their audience.”

Why is fake news so popular on newssites? Here are two reasons: first, it provides emotional “buzz.” Second, because it can make a lot of money. As the Washington Bureau chief of the Pulitzer winning Huffington Post lamented, “If you throw something up without fact checking it, and you’re the first one to put it up, and you get millions and millions of views, and later it’s proved false, you still got those views. That’s a problem. The incentives are all wrong.”Especially when, as at places like Bloomberg, remuneration is based on the number of hits an article receives. But incentives are wrong not just for news gathering organizations.

Americans continue to believe important historical “facts” that are untrue. After 9/11, Americans were incensed to hear that the many in the Middle East thought Osama bin Laden’s horrifying attack was the product of a CIA-Mossad plot. To this day, large majorities in countries like Pakistan think the massacre of 9/11 was created in Washington or Tel Aviv.

Yet close to a majority of Americans believe that Saddam Hussein, tyrant of Iraq, was in cahoots with Al-Qaeda, especially before the 9/11 attack. The Bush administration told them so.Which people in the Middle East rightly regard as preposterous.Saddam Hussein was the leader of a boldly secular, Arabist tyranny. Sunni fanatics like Al Qaeda were his regime’s blood enemies. That they would work together rather than murder each other was just insane. Welcome to the world of fict and faction.

What can we learn from this? Plausibility is not truth; when something is “too good to be true” it generally isn’t; institutions increasingly do not back up what they proclaim and sell.And the “free informational marketplace” of the Internet is a wonderful site for fraud, scams, lies, plausible lies, and pleasant, beautiful untruths. So we all need our own truth detectors.

In the end, you realize your only real option – against almost every fiber in your irate being – is to take each knot in turn, patiently and gently undo it, loosen a little, see what happens, and move on to the next. You will never know exactly when all the knots will resolve themselves – it can happen quite quickly after a while or seemingly never. But you do know that patience, and concern with the here and now, is the only way to “solve” the “problem.” You don’t look forward with a plan; you look down with a practice.

we can say what God is not, we can speak of his attributes, but we cannot say what He is. That apophatic dimension, which reveals how I speak about God, is critical to our theology

I would also classify as arrogant those theologies that not only attempted to define with certainty and exactness God’s attributes, but also had the pretense of saying who He was.

It is only in living that we achieve hints and guesses – and only hints and guesses – of what the Divine truly is. And because the Divine is found and lost by humans in time and history, there is no reachable truth for humans outside that time and history.

We are part of an unfolding drama in which the Christian, far from clinging to some distant, pristine Truth he cannot fully understand, will seek to understand and discern the “signs of the times” as one clue as to how to live now, in the footsteps of Jesus. Or in the words of T.S. Eliot, There is only the fight to recover what has been lost And found and lost again and again: and now, under conditions That seem unpropitious. But perhaps neither gain nor loss. For us, there is only the trying. The rest is not our business.

Ratzinger’s Augustinian notion of divine revelation: it is always a radical gift; it must always be accepted without question; it comes from above to those utterly unworthy below; and we are too flawed, too sinful, too human to question it in even the slightest respect. And if we ever compromise an iota on that absolute, authentic, top-down truth, then we can know nothing as true. We are, in fact, lost for ever.

A Christian life is about patience, about the present and about trust that God is there for us. It does not seek certainty or finality to life’s endless ordeals and puzzles. It seeks through prayer and action in the world to listen to God’s plan and follow its always-unfolding intimations. It requires waiting. It requires diligence

We may never know why exactly Benedict resigned as he did. But I suspect mere exhaustion of the body and mind was not the whole of it. He had to see, because his remains such a first-rate mind, that his project had failed, that the levers he continued to pull – more and more insistent doctrinal orthodoxy, more political conflict with almost every aspect of the modern world, more fastidious control of liturgy – simply had no impact any more.

The Pope must accompany those challenging existing ways of doing things! Others may know better than he does. Or, to feminize away the patriarchy: I dream of a church that is a mother and shepherdess. The church’s ministers must be merciful, take responsibility for the people, and accompany them like the good Samaritan, who washes, cleans, and raises up his neighbor. This is pure Gospel.

the key to Francis’ expression of faith is an openness to the future, a firm place in the present, and a willingness to entertain doubt, to discern new truths and directions, and to grow. Think of Benedict’s insistence on submission of intellect and will to the only authentic truth (the Pope’s), and then read this: Within the Church countless issues are being studied and reflected upon with great freedom. Differing currents of thought in philosophy, theology, and pastoral practice, if open to being reconciled by the Spirit in respect and love, can enable the Church to grow, since all of them help to express more clearly the immense riches of God’s word. For those who long for a monolithic body of doctrine guarded by all and leaving no room for nuance, this might appear as undesirable and leading to confusion. But in fact such variety serves to bring out and develop different facets of the inexhaustible riches of the Gospel.

Francis, like Jesus, has had such an impact in such a short period of time simply because of the way he seems to be. His being does not rely on any claims to inherited, ecclesiastical authority; his very way of life is the only moral authority he wants to claim.

faith is, for Francis, a way of life, not a set of propositions. It is a way of life in community with others, lived in the present yet always, deeply, insistently aware of eternity.

Father Howard Gray S.J. has put it simply enough: Ultimately, Ignatian spirituality trusts the world as a place where God dwells and labors and gathers all to himself in an act of forgiveness where that is needed, and in an act of blessing where that is prayed for.

Underlying all this is a profound shift away from an idea of religion as doctrine and toward an idea of religion as a way of life. Faith is a constantly growing garden, not a permanently finished masterpiece

Some have suggested that much of what Francis did is compatible with PTSD. He disowned his father and family business, and he chose to live homeless, and close to naked, in the neighboring countryside, among the sick and the animals. From being the dashing man of society he had once been, he became a homeless person with what many of us today would call, at first blush, obvious mental illness.

these actions – of humility, of kindness, of compassion, and of service – are integral to Francis’ resuscitation of Christian moral authority. He is telling us that Christianity, before it is anything else, is a way of life, an orientation toward the whole, a living commitment to God through others. And he is telling us that nothing – nothing – is more powerful than this.

I would not speak about, not even for those who believe, an “absolute” truth, in the sense that absolute is something detached, something lacking any relationship. Now, the truth is a relationship! This is so true that each of us sees the truth and expresses it, starting from oneself: from one’s history and culture, from the situation in which one lives, etc. This does not mean that the truth is variable and subjective. It means that it is given to us only as a way and a life. Was it not Jesus himself who said: “I am the way, the truth, the life”? In other words, the truth is one with love, it requires humbleness and the willingness to be sought, listened to and expressed.

“proselytism is solemn nonsense.” That phrase – deployed by the Pope in dialogue with the Italian atheist Eugenio Scalfari (as reported by Scalfari) – may seem shocking at first. But it is not about denying the revelation of Jesus. It is about how that revelation is expressed and lived. Evangelism, for Francis, is emphatically not about informing others about the superiority of your own worldview and converting them to it. That kind of proselytism rests on a form of disrespect for another human being. Something else is needed:

nstead of seeming to impose new obligations, Christians should appear as people who wish to share their joy, who point to a horizon of beauty and who invite others to a delicious banquet. It is not by proselytizing that the Church grows, but “by attraction.”

what you see in the life of Saint Francis is a turn from extreme violence to extreme poverty, as if only the latter could fully compensate for the reality of the former. This was not merely an injunction to serve the poor. It is the belief that it is only by being poor or becoming poor that we can come close to God

Pope Francis insists – and has insisted throughout his long career in the church – that poverty is a key to salvation. And in choosing the name Francis, he explained last March in Assisi, this was the central reason why:

Saint Francis. His conversion came after he had gone off to war in defense of his hometown, and, after witnessing horrifying carnage, became a prisoner of war. After his release from captivity, his strange, mystical journey began.

the priority of practice over theory, of life over dogma. Evangelization is about sitting down with anyone anywhere and listening and sharing and being together. A Christian need not be afraid of this encounter. Neither should an atheist. We are in this together, in the same journey of life, with the same ultimate mystery beyond us. When we start from that place – of radical humility and radical epistemological doubt – proselytism does indeed seem like nonsense, a form of arrogance and detachment, reaching for power, not freedom. And evangelization is not about getting others to submit their intellect and will to some new set of truths; it is about an infectious joy for a new way of living in the world. All it requires – apart from joy and faith – is patience.

“Preach the Gospel always. If necessary, with words.”

But there is little sense that a political or economic system can somehow end the problem of poverty in Francis’ worldview. And there is the discomfiting idea that poverty itself is not an unmitigated evil. There is, indeed, a deep and mysterious view, enunciated by Jesus, and held most tenaciously by Saint Francis, that all wealth, all comfort, and all material goods are suspect and that poverty itself is a kind of holy state to which we should all aspire.

Not only was Saint Francis to become homeless and give up his patrimony, he was to travel on foot, wearing nothing but a rough tunic held together with rope. Whatever else it is, this is not progressivism. It sees no structural, human-devised system as a permanent improver of our material lot. It does not envision a world without poverty, but instead a church of the poor and for the poor. The only material thing it asks of the world, or of God, is daily bread – and only for today, never for tomorrow.

From this perspective, the idea that a society should be judged by the amount of things it can distribute to as many people as possible is anathema. The idea that there is a serious social and political crisis if we cannot keep our wealth growing every year above a certain rate is an absurdity.

this is a 21st-century heresy. Which means, I think, that this Pope is already emerging and will likely only further emerge as the most potent critic of the newly empowered global capitalist project.

Now, the only dominant ideology in the world is the ideology of material gain – either through the relatively free markets of the West or the state-controlled markets of the East. And so the church’s message is now harder to obscure. It stands squarely against the entire dominant ethos of our age. It is the final resistance.

For Francis, history has not come to an end, and capitalism, in as much as it is a global ideology that reduces all of human activity to the cold currency of wealth, is simply another “ism” to be toppled in humankind’s unfolding journey toward salvation on earth.

Francis will grow as the church reacts to him; it will be a dynamic, not a dogma; and it will be marked less by the revelation of new things than by the new recognition of old things, in a new language. It will be, if its propitious beginnings are any sign, a patient untying of our collective, life-denying knots.

What would college admission look like if we flipped this whole understanding and placed a premium on empathy or the ability to recognize and create things of great beauty

What was wrong with their education? In Wiesel’s words: "It emphasized theories instead of values, concepts rather than human beings, abstraction rather than consciousness, answers instead of questions, ideology and efficiency rather than conscience."

There is an information explosion going on, by which I mean a rapid increase of data, words, and paper. But this explosion should not be taken for an increase in knowledge and wisdom, which cannot so easily by measured. What can be said truthfully is that some knowledge is increasing while other kinds of knowledge are being lost. David Ehrenfeld has pointed out that biology departments no longer hire faculty in such areas as systematics, taxonomy, or ornithology. In other words, important knowledge is being lost because of the recent overemphasis on molecular biology and genetic engineering, which are more lucrative, but not more important, areas of inquiry.

In the modern curriculum we have fragmented the world into bits and pieces called disciplines and subdisciplines. As a result, after 12 or 16 or 20 years of education, most students graduate without any broad integrated sense of the unity of things. The consequences for their personhood and for the planet are large. For example, we routinely produce economists who lack the most rudimentary knowledge of ecology. This explains why our national accounting systems do not subtract the costs of biotic impoverishment, soil erosion, poisons in the air or water, and resource depletion from gross national product. We add the price of the sale of a bushel of wheat to GNP while forgetting to subtract the three bushels of topsoil lost in its production.

The plain fact is that the planet does not need more "successful" people. But it does desperately need more peacemakers, healers, restorers, storytellers, and lovers of every shape and form. It needs people who live well in their places.

The goal of education is not mastery of subject matter, but of one’s person. Subject matter is simply the tool. Much as one would use a hammer and chisel to carve a block of marble, one uses ideas and knowledge to forge one’s own personhood.

knowledge carries with it the responsibility to see that it is well used in the world.

we cannot say that we know something until we understand the effects of this knowledge on real people and their communities

Indoor classes create the illusion that learning only occurs inside four walls isolated from what students call without apparent irony the "real world."

The true work of the mathematician is not experienced until the later parts of graduate school, when the student is challenged to create knowledge in the form of a novel proof. It is common to fill page after page with an attempt, the seasons turning, only to arrive precisely where you began, empty-handed — or to realize that a subtle flaw of logic doomed the whole enterprise from its outset. The steady state of mathematical research is to be completely stuck. It is a process that Charles Fefferman of Princeton, himself a onetime math prodigy turned Fields medalist, likens to ‘‘playing chess with the devil.’’ The rules of the devil’s game are special, though: The devil is vastly superior at chess, but, Fefferman explained, you may take back as many moves as you like, and the devil may not. You play a first game, and, of course, ‘‘he crushes you.’’ So you take back moves and try something different, and he crushes you again, ‘‘in much the same way.’’ If you are sufficiently wily, you will eventually discover a move that forces the devil to shift strategy; you still lose, but — aha! — you have your first clue.

Tao has emerged as one of the field’s great bridge-­builders. At the time of his Fields Medal, he had already made discoveries with more than 30 different collaborators. Since then, he has also become a prolific math blogger with a decidedly non-­Gaussian ebullience: He celebrates the work of others, shares favorite tricks, documents his progress and delights at any corrections that follow in the comments. He has organized cooperative online efforts to work on problems. ‘‘Terry is what a great 21st-­century mathematician looks like,’’ Jordan Ellenberg, a mathematician at the University of Wisconsin, Madison, who has collaborated with Tao, told me. He is ‘‘part of a network, always communicating, always connecting what he is doing with what other people are doing.’’

As a group, the people drawn to mathematics tend to value certainty and logic and a neatness of outcome, so this game becomes a special kind of torture. And yet this is what any ­would-be mathematician must summon the courage to face down: weeks, months, years on a problem that may or may not even be possible to unlock. You find yourself sitting in a room without doors or windows, and you can shout and carry on all you want, but no one is listening.

For their work, Tao and Green salvaged a crucial bit from an earlier proof done by others, which had been discarded as incorrect, and aimed at a different goal. Other maneuvers came from masterful proofs by Timothy Gowers of England and Endre Szemeredi of Hungary. Their work, in turn, relied on contributions from Erdos, Klaus Roth and Frank Ramsey, an Englishman who died at age 26 in 1930, and on and on, into history. Ask mathematicians about their experience of the craft, and most will talk about an intense feeling of intellectual camaraderie. ‘‘A very central part of any mathematician’s life is this sense of connection to other minds, alive today and going back to Pythagoras,’’ said Steven Strogatz, a professor of mathematics at Cornell University. ‘‘We are having this conversation with each other going over the millennia.’’

Most mathematicians tend to specialize, but Tao ranges widely, learning from others and then working with them to make discoveries. Markus Keel, a longtime collaborator and close friend, reaches to science fiction to explain Tao’s ability to rapidly digest and employ mathematical ideas: Seeing Tao in action, Keel told me, reminds him of the scene in ‘‘The Matrix’’ when Neo has martial arts downloaded into his brain and then, opening his eyes, declares, ‘‘I know kung fu.’’ The citation for Tao’s Fields Medal, awarded in 2006, is a litany of boundary hopping and notes particularly ‘‘beautiful work’’ on Horn’s conjecture, which Tao completed with a friend he had played foosball with in graduate school. It was a new area of mathematics for Tao, at a great remove from his known stamping grounds. ‘‘This is akin,’’ the citation read, ‘‘to a leading English-­language novelist suddenly producing the definitive Russian novel.’’

An effort to prove that 1 equals 0 is not likely to yield much fruit, it’s true, but the hacker’s mind-set can be extremely useful when doing math. Long ago, mathematicians invented a number that when multiplied by itself equals negative 1, an idea that seemed to break the basic rules of multiplication. It was so far outside what mathematicians were doing at the time that they called it ‘‘imaginary.’’ Yet imaginary numbers proved a powerful invention, and modern physics and engineering could not function without them.

Early encounters with math can be misleading. The subject seems to be about learning rules — how and when to apply ancient tricks to arrive at an answer. Four cookies remain in the cookie jar; the ball moves at 12.5 feet per second. Really, though, to be a mathematician is to experiment. Mathematical research is a fundamentally creative act. Lore has it that when David Hilbert, arguably the most influential mathematician of fin de siècle Europe, heard that a colleague had left to pursue fiction, he quipped: ‘‘He did not have enough imagination for mathematics.’’

Many people think that substantial progress on Navier-­Stokes may be impossible, and years ago, Tao told me, he wrote a blog post concurring with this view. Now he has some small bit of hope. The twin-prime conjecture had the same feel, a sense of breaking through the wall of intimidation that has scared off many aspirants. Outside the world of mathematics, both Navier-­Stokes and the twin-prime conjecture are described as problems. But for Tao and others in the field, they are more like opponents. Tao’s opponent has been known to taunt him, convincing him that he is overlooking the obvious, or to fight back, making quick escapes when none should be possible. Now the opponent appears to have revealed a weakness. But Tao said he has been here before, thinking he has found a way through the defenses, when in fact he was being led into an ambush. ‘‘You learn to get suspicious,’’ Tao said. ‘‘You learn to be on the lookout.’’

But the students also described a sense of loss.

A 15-year-old boy told me that someday he wanted to raise a family, not the way his parents are raising him (with phones out during meals and in the park and during his school sports events) but the way his parents think they are raising him — with no phones at meals and plentiful family conversation. One college junior tried to capture what is wrong about life in his generation. “Our texts are fine,” he said. “It’s what texting does to our conversations when we are together that’s the problem.”

One teacher observed that the students “sit in the dining hall and look at their phones. When they share things together, what they are sharing is what is on their phones.” Is this the new conversation? If so, it is not doing the work of the old conversation. The old conversation taught empathy. These students seem to understand each other less.

In 2010, a team at the University of Michigan led by the psychologist Sara Konrath put together the findings of 72 studies that were conducted over a 30-year period. They found a 40 percent decline in empathy among college students, with most of the decline taking place after 2000.

We’ve gotten used to being connected all the time, but we have found ways around conversation — at least from conversation that is open-ended and spontaneous, in which we play with ideas and allow ourselves to be fully present and vulnerable. But it is in this type of conversation — where we learn to make eye contact, to become aware of another person’s posture and tone, to comfort one another and respectfully challenge one another — that empathy and intimacy flourish. In these conversations, we learn who we are.

the trend line is clear. It’s not only that we turn away from talking face to face to chat online. It’s that we don’t allow these conversations to happen in the first place because we keep our phones in the landscape.

It’s a powerful insight. Studies of conversation both in the laboratory and in natural settings show that when two people are talking, the mere presence of a phone on a table between them or in the periphery of their vision changes both what they talk about and the degree of connection they feel. People keep the conversation on topics where they won’t mind being interrupted. They don’t feel as invested in each other. Even a silent phone disconnects us.

we have put this virtuous circle in peril. We turn time alone into a problem that needs to be solved with technology.

At a nightly cabin chat, a group of 14-year-old boys spoke about a recent three-day wilderness hike. Not that many years ago, the most exciting aspect of that hike might have been the idea of roughing it or the beauty of unspoiled nature. These days, what made the biggest impression was being phoneless. One boy called it “time where you have nothing to do but think quietly and talk to your friends.” The campers also spoke about their new taste for life away from the online feed. Their embrace of the virtue of disconnection suggests a crucial connection: The capacity for empathic conversation goes hand in hand with the capacity for solitude.

In solitude we find ourselves; we prepare ourselves to come to conversation with something to say that is authentic, ours. If we can’t gather ourselves, we can’t recognize other people for who they are. If we are not content to be alone, we turn others into the people we need them to be. If we don’t know how to be alone, we’ll only know how to be lonely.

Yalda T. Uhls was the lead author on a 2014 study of children at a device-free outdoor camp. After five days without phones or tablets, these campers were able to read facial emotions and correctly identify the emotions of actors in videotaped scenes significantly better than a control group. What fostered these new empathic responses? They talked to one another. In conversation, things go best if you pay close attention and learn how to put yourself in someone else’s shoes. This is easier to do without your phone in hand. Conversation is the most human and humanizing thing that we do.

People sometimes say to me that they can see how one might be disturbed when people turn to their phones when they are together. But surely there is no harm when people turn to their phones when they are by themselves? If anything, it’s our new form of being together.

But this way of dividing things up misses the essential connection between solitude and conversation. In solitude we learn to concentrate and imagine, to listen to ourselves. We need these skills to be fully present in conversation.

One start toward reclaiming conversation is to reclaim solitude. Some of the most crucial conversations you will ever have will be with yourself. Slow down sufficiently to make this possible. And make a practice of doing one thing at a time. Think of unitasking as the next big thing. In every domain of life, it will increase performance and decrease stress.

Multitasking comes with its own high, but when we chase after this feeling, we pursue an illusion. Conversation is a human way to practice unitasking.

Our phones are not accessories, but psychologically potent devices that change not just what we do but who we are. A second path toward conversation involves recognizing the degree to which we are vulnerable to all that connection offers. We have to commit ourselves to designing our products and our lives to take that vulnerability into account.

We can choose not to carry our phones all the time. We can park our phones in a room and go to them every hour or two while we work on other things or talk to other people. We can carve out spaces at home or work that are device-free, sacred spaces for the paired virtues of conversation and solitude.

Families can find these spaces in the day to day — no devices at dinner, in the kitchen and in the car.

Engineers are ready with more ideas: What if our phones were not designed to keep us attached, but to do a task and then release us? What if the communications industry began to measure the success of devices not by how much time consumers spend on them but by whether it is time well spent?

The young woman who is so clear about the seven minutes that it takes to see where a conversation is going admits that she often doesn’t have the patience to wait for anything near that kind of time before going to her phone. In this she is characteristic of what the psychologists Howard Gardner and Katie Davis called the “app generation,” which grew up with phones in hand and apps at the ready. It tends toward impatience, expecting the world to respond like an app, quickly and efficiently. The app way of thinking starts with the idea that actions in the world will work like algorithms: Certain actions will lead to predictable results.

This attitude can show up in friendship as a lack of empathy. Friendships become things to manage; you have a lot of them, and you come to them with tools

here is a first step: To reclaim conversation for yourself, your friendships and society, push back against viewing the world as one giant app. It works the other way, too: Conversation is the antidote to the algorithmic way of looking at life because it teaches you about fluidity, contingency and personality.

We have time to make corrections and remember who we are — creatures of history, of deep psychology, of complex relationships, of conversations, artless, risky and face to face.

Can grammatical gender influence speakers’ cognitive processes when they’re speaking another language entirely?

They created a list of 24 objects that have opposite genders in Spanish and German; in each language, half of the objects were masculine and half were feminine.

peaking English and using materials written in English

The word “bridge” is feminine in German and masculine in Spanish. Sure enough, German speakers described bridges as beautiful, elegant, fragile, pretty, and slender, while Spanish speakers said they were big, dangerous, strong, sturdy, and towering.

As these studies show, grammatical gender can influence people’s thinking, even when they’re speaking a language with no grammatical gender to speak of —and even when they’re not speaking any language at all!

Small, unnoticed features of language can influence our thoughts, sometimes in big ways. Knowing that, imagine how else language, culture, and society might affect our thoughts, feelings, and behaviors

it seems we are not yet free of gender stereotypes

cultural memes. The Darwinesque concept originates in Dawkins's 1976 book, The Selfish Gene. Memes are analogous to genes, Dennett has said, "replicating units of culture" that spread from mind to mind like a virus. Religion, chess, songs, clothing, tolerance for free speech—all have been described as memes. Tallis considers it absurd to talk of a noun-phrase like "tolerance for free speech" as a discrete entity. But Dennett argues that Tallis's objections are based on "a simplistic idea of what one might mean by a unit." Memes aren't units? Well, in that spirit, says Dennett, organisms aren't units of biology, nor are species—they're too complex, with too much variation. "He's got to allow theory to talk about entities which are not simple building blocks," Dennett says.

Geraint Rees, director of University College London's Institute of Cognitive Neuroscience, complains that reading Tallis is "a bit like trying to nail jelly to the wall." He "rubbishes every current theory of the relationship between mind and brain, whether philosophical or neuroscientific," while offering "little or no alternative,"

How is it that he perceives the glass of water on the table? How is it that he feels a sense of self over time? How is it that he can remember a patient he saw in 1973, and then cast his mind forward to his impending visit to the zoo? There are serious problems with trying to reduce such things to impulses in the brain, he argues. We can explain "how the light gets in," he says, but not "how the gaze looks out." And isn't it astonishing, he adds, that much neural activity seems to have no link to consciousness? Instead, it's associated with things like controlling automatic movements and regulating blood pressure. Sure, we need the brain for consciousness: "Chop my head off, and my IQ descends." But it's not the whole story. There is more to perceptions, memories, and beliefs than neural impulses can explain. The human sphere encompasses a "community of minds," Tallis has written, "woven out of a trillion cognitive handshakes of shared attention, within which our freedom operates and our narrated lives are led." Those views on perception and memory anchor his attack on "neurobollocks." Because if you can't get the basics right, he says, then it's premature to look to neuroscience for clues to complex things like love.

Yes, many unanswered questions persist. But these are early days, and neuroscience remains immature, says Churchland, a professor emerita of philosophy at University of California at San Diego and author of the subfield-spawning 1986 book Neurophilosophy. In the 19th century, she points out, people thought we'd never understand light. "Well, by gosh," she says, "by the time the 20th century rolls around, it turns out that light is electromagnetic radiation. ... So the fact that at a certain point in time something seems like a smooth-walled mystery that we can't get a grip on, doesn't tell us anything about whether some real smart graduate student is going to sort it out in the next 10 years or not."

Dennett claims he's got much of it sorted out already. He wrote a landmark book on the topic in 1991, Consciousness Explained. (The title "should have landed him in court, charged with breach of the Trade Descriptions Act," writes Tallis.) Dennett uses the vocabulary of computer science to explain how consciousness emerges from the huge volume of things happening in the brain all at once. We're not aware of everything, he tells me, only a "limited window." He describes that stream of consciousness as "the activities of a virtual machine which is running on the parallel hardware of the brain." "You—the fruits of all your experience, not just your genetic background, but everything you've learned and done and all your memories—what ties those all together? What makes a self?" Dennett asks. "The answer is, and has to be, the self is like a software program that organizes the activities of the brain."

Behaviors strongly associated with depression—reduced physical activity and human contact, overconsumption of processed food, seeking endless distraction—are the very behaviors that more and more people now can do, are even forced to do by the nature of their sedentary, indoor jobs.

e we are gathering scientific evidence for the benefits of living close to nature, not simply for enjoying its beauty or getting spiritual sustenance but for keeping our brains and nervous systems in good working order.

Human beings evolved to thrive in natural environments and in bonded social groups. Few of us today can enjoy such a life and the emotional equilibrium it engenders, but our genetic predisposition for it has not changed.

I believ

It may seem baffling, but the explanation is simple: the human body was never designed for the modern postindustrial environment.”

Possibly, the deterioration of emotional well-being characteristic of contemporary urban life represents a cumulative effect of lifestyle changes that have been occurring over many years, an effect that is now suddenly obvious.

Not only do we suffer from nature deficit, we are experiencing information surfeit. Many people today spend much of their waking time surfing the Internet, texting and talking on mobile phones, attending to email, watching television, and being stimulated by other new media—experiences never available until now.

The allure of synthetic entertainment—television, the Internet—is eerily reminiscent of the false promise of industrial food. It seems like a distillation of the good aspects of a social life, always entertaining yet easy to abandon when it becomes tedious or challenging. But, like junk food, it is ultimately unsatisfying and potentially harmful. Our brains, genetically adapted to help us negotiate a successful course through complex, changing, and often hazardous natural environments, are suddenly confronted with an overload of information and stimulation independent of physical reality.