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I saw an older woman, hair neatly coifed, beat a policeman with her handbag. A young father hoisted his little girl on his shoulder, carefully affixing a surgical mask to her face, an imperfect shield from the gas. A student in Kwangju bit his finger and wrote protest slogans in his own blood.

At the trial, his father lunged at the three policemen, small and scared now, protected by more than 50 guards. Screams broke out and a purse flew through the air at the judges as the light sentence was read. Women whose sons were still in jail stormed the bus carrying the policemen, throwing bottles against the windows. Plainclothes police shoved the four of them onto the concrete, where they lay unconscious.

Laws are still on the books that can be used as tools to stifle dissent. South Korea remains shadowed by legitimate fears of North Korean aggression and espionage, but those were and are exploited by the government.

This is not a tame society, for all the comforts its public has won in the years since. This may be the land of Psy and Gangnam style, a country so wired that some of its children are sent to boot camps to wean them from internet addiction.

Politicians buck the popular will at their peril.

You similarly don’t want to come crashing down on your knee so you break your kneecap or do that maneuver where you kind of pedal with your feet to catch yourself, which can lead to broken bones in your foot and ankle.

Paratroopers’ goal is to fall sideways in the direction the wind is carrying them — in no way resisting the momentum of the fall. When the balls of their feet barely reach the ground, they immediately distribute the impact in rapid sequence up through the calf to the thigh and buttocks.

Accept that you’re falling and go with it, round your body, and don’t stiffen and distribute the energy so you take the fall in the widest area possible,

Young children are arguably the best fallers because they have yet to develop fear or embarrassment, so they just tumble and roll without tensing up and trying to catch themselves.

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.’’

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.’’

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.’’

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.

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.’’

In 2014, Wynn noticed a trend on YouTube that disturbed her: Videos with hyperbolic titles like “why feminism ruins everything,” “SJW cringe compilation,” and “Ben Shapiro DESTROYS Every College Snowflake” were attracting millions of views and spawning long, jeering comment threads. Wynn felt she was watching the growth of a community of outrage that believes feminists, Marxists, and multiculturalists are conspiring to destroy freedom of speech, liquidate gender norms, and demolish Western civilization

Wynn created ContraPoints to offer entertaining, coherent rebuttals to these kinds of ideas. Her videos also explain left-wing talking points—like rape culture and cultural appropriation—and use philosophy to explore topics that are important to Wynn, such as the meaning of gender for trans people.

Wynn thinks it’s a mistake to assume that viewers of angry, right-wing videos are beyond redemption. “It’s quite difficult to get through to the people who are really committed to these anti-progressive beliefs,” Wynn told me recently. However, she said, she believes that many viewers find such ideas “psychologically resonant” without being hardened reactionaries. This broad, not fully committed center—comprising people whose minds can still be changed—is Wynn’s target audience.

Usually, the videos to which Wynn is responding take the stance of dogged reason cutting through the emotional excesses of so-called “political correctness.” For example, the American conservative commentator Ben Shapiro, who is a target of a recent ContraPoints video, has made “facts don’t care about your feelings” his motto. Wynn’s first step in trying to win over those who find anti-progressive views appealing is to show that these ideas often rest on a flimsy foundation. To do so, she fully adopts the rational standards of argument that her rivals pride themselves on following, and demonstrates how they fail to achieve them

Wynn dissects her opponents’ positions, holding up fallacies, evasions, and other rhetorical tricks for examination, all the while providing a running commentary on good argumentative method.

The host defends her own positions according to the same principles. Wynn takes on the strongest version of her opponent’s argument, acknowledges when she thinks her opponents are right and when she has been wrong, clarifies when misunderstood, and provides plenty of evidence for her claims

Wynn is a former Ph.D. student in philosophy, and though her videos are too rich with dick jokes for official settings, her argumentative practice would pass muster in any grad seminar.

she critiques many of her leftist allies for being bad at persuasion.

Socrates persuaded by both the logic of argument and the dynamic of fandom. Wynn is beginning to grow a dedicated following of her own: Members of online discussion groups refer to her as “mother” and “the queen,” produce fan art, and post photos of themselves dressed as characters from her videos.

she shares Socrates’s view that philosophy is more an erotic art than a martial one

As she puts it, she’s not trying to destroy the people she addresses, but seduce them

for Wynn, the true key to persuasion is to engage her audience on an emotional level.

One thing she has come across repeatedly is a disdain for the left’s perceived moral superiority. Anti-progressives of all stripes, Wynn told me, show an “intense defensiveness against being told what to do” and a “repulsion in response to moralizing.”

Matching her speech to the audience’s tastes presents a prickly rhetorical challenge. In an early video, Contra complains: “The problem is this medium. These goddamn savages demand a circus, and I intend to give them one, but behind the curtain, I really just want to have a conversation.

Philosophical conversation requires empathy and good-faith engagement. But the native tongue of political YouTube is ironic antagonism. It’s Wynn’s inimitable way of combining these two ingredients that gives ContraPoints its distinctive mouthfeel.

Wynn spends weeks in the online communities of her opponents—whether they’re climate skeptics or trans-exclusionary feminists—trying to understand what they believe and why they believe it. In Socrates’s words, she’s studying the souls of her audience.

Emotions should never be bottled up. Call or go see someone you trust and recount to them what happened. Hearing an opinion other than your own broadens your awareness. Keep a journal and transfer your emotions from your inner self onto the paper. Many people find it helpful to engage in aggressive exercises, such as kickboxing or martial arts, to discharge their feelings.

Every happening of our lives, whether good or bad, serves a higher purpose. Wisdom means being able to see past the moment and discern the greater meaning of any given situation. You may not understand it in the beginning, but as time goes by, you'll begin to see the bigger picture falling into perfect order.

Whenever you are confronted with an emotion which is making you feel or think something bad, force it out of your mind and replace it with a different thought.

A constant reminder of our ardent nature, emotions surge through us at every second of the day.