Last week, Pete Buttigieg, the young mayor of South Bend, Indiana, who recently announced that he is exploring a Presidential candidacy, arrived in New York to meet the press. First up, on Thursday, was an interview on “CBS This Morning,” where the show’s hosts seemed slightly impatient, like college-admissions officers who had been asked to interview a benefactor’s son. Norah O’Donnell positioned her eyebrows skeptically. “You’re thirty-seven, you represent a town of a hundred and two thousand people—did I get that right? What qualifies you to be President of the United States?” Buttigieg, who has pale skin, thick brown hair, and a formal manner, gave a self-deprecating laugh. “I know that I’m the youngest person in this conversation, but I think that the experience of leading a city through a transformation is really relevant right now,” he said. “Things are changing tectonically in our country, and we can’t just keep doing what we’ve been doing. We can’t nibble around the edges of a system that no longer works.” John Dickerson pointed out that other Democratic candidates were proposing very big ideas—Medicare for All, the abolition of private health insurance—and asked, “What is your idea that is so big that nobody would mistake it for nibbling around the edges?” Buttigieg answered, “Well, first of all, we’ve got to repair our democracy. The Electoral College needs to go, because it’s made our society less and less democratic.” He went on in this vein, suggesting that electoral reform was essential, and promising that other policies, on security and health care, would follow. Viewers were left with the image of an impressive and fluent young politician, whose presence in the Presidential race, and on their screens, had never really been explained.

A few hours later, I met Buttigieg in a busy restaurant in the basement of Rockefeller Center, where the windows looked out at the ice-skating rink. He had taken off his sports coat for an appearance on “The View,” but put it back on for lunch, and he arrived carrying an enormous backpack over his left shoulder. “The View” had gone much better. The hosts were intrigued by the idea that Buttigieg, who came out three and a half years ago, could be the first gay President, and by his campaign’s main theme, which he calls intergenerational justice—he believes that millennials are suffering from their elders’ short-term thinking on climate change, economics, and other issues. Whoopi Goldberg wondered whether such a case could be made without alienating older Americans, and Buttigieg watched her intently, absorbing the criticism. “I think we really hit on something with this idea of intergenerational justice,” Buttigieg told me. “I think the trick for us—and this was a big part of what Whoopi Goldberg was asking about—is there should be a way to make a generational case without this all being about generational conflict. And I think there’s a way to do it.”

Buttigieg, who attended Harvard, studied philosophy, politics, and economics (P.P.E.) at Oxford on a Rhodes Scholarship, and did a tour in Afghanistan as a naval reservist, can seem like an “old person’s idea of a young person,” as Michael Kinsley once said of Al Gore. Certainly, against the image of the millennial left, and of Alexandria Ocasio-Cortez, Buttigieg appears to be a more prosaic political character—he has a habit of giving answers in numbered sequence, and he uses phrases like “pathway to peace.” But, in his own understated way, he is suggesting a sharp break with the past. If you thought in terms of the effects of public policy on millennials, he said, you began to see generational imbalances everywhere. The victims of school shootings suffered because of the gun liberties given to older Americans. Cutting taxes for the richest Americans meant that young people, inevitably, would have to pay the bill. Climate policy, he said, was the deepest example of the imbalance, but the Iraq War was perhaps the most tangible. “There’s this romantic idea that’s built up around war,” he said. “But the pragmatic view is there are tons of people of my generation who have lost their lives, lost their marriages, or lost their health as a consequence of being sent to wars which could have been avoided.” Then he quoted, happily, from “Lawrence of Arabia”: “The virtues of war are the virtues of young men—courage and hope for the future. The vices of peace are the vices of old men—mistrust and caution.”

For much of his life, Buttigieg has been giving those around him the impression of extreme promise. Both of his parents were professors at Notre Dame, and he grew up in South Bend, near the campus. His father, Joe, was a translator of the Italian Marxist theorist Antonio Gramsci and a scholar of James Joyce. His mother, Anne Montgomery, is a linguist. At Harvard, Buttigieg was the student president of the Institute of Politics, a role sought by the most ambitious of the exceptionally ambitious, but he could also suggest a more inquisitive nature. His close friend Nathaniel Myers recalled that Buttigieg had become entranced by the Norwegian novel “Naïve. Super,” by Erlend Loe, taught himself the language to translate another work by the author, and then started periodically attending a Norwegian church in Chicago to keep up. He plays piano, and has sat in with the South Bend Symphony Orchestra and Ben Folds. He was elected mayor of South Bend, in 2011, when he was twenty-nine, and only came out in advance of his reëlection campaign, when he was thirty-three. His wedding, to Chasten Glezman, who was a Montessori middle-school teacher, was broadcast live online.

In 2015, Buttigieg gave a speech at Harvard, and David Axelrod, President Obama’s longtime chief strategist, was in the audience. The speech, Axelrod told me this week, was moving and thoughtful, and he noticed that, though Buttigieg had notes, he rarely consulted them. What struck him was a familiar kind of talent. “His story is an incredible story,” Axelrod said, “but more impressive than the story is the guy. At a time when people are aching for hope and a path forward that we can all walk, he is a relentlessly positive person.”

The following year, Frank Bruni wrote a column proposing Buttigieg as “the first gay President.” In an interview with David Remnick, Obama included Buttigieg on a short list of gifted rising Democrats. “If I told you he was anything other than a long shot, you’d hang up the phone,” Axelrod said, of the Presidential race, but he emphasized the possibility that, as he put it, lightning could strike. “The practical political point is it’s hard to see where he’s going in Indiana. If it doesn’t work out, if there’s a Democratic President looking for talent, I know Pete well enough to know he’s going to be high on the list, and higher for having run.” At the very least, Axelrod said, Buttigieg was likely to emerge from this as “an interesting voice from his generation.”

Part of the paradox of Buttigieg’s candidacy is that he has placed himself in a performative role, without the benefit of a performative personality. “He is reserved, and maybe that’s a hindrance,” Axelrod told me. Chasten Glezman, his husband, told a reporter that Buttigieg is “still coming out of some shells.” In our conversation, he seemed most practiced when talking about policy but most alive when discussing James Joyce. When I asked how he had made the decision to run for President, he brightened, and said that, though he wasn’t a Catholic, he made use of the “Ignatian process of discernment.” He pictured a world in which he became President—perhaps shy of using the word, he referred to it only as “the end state”—and then considered whether it gave him a feeling of “fulfillment or desolation.” Fulfillment, it turned out.

I had noticed that, in his interview on “CBS This Morning,” no one mentioned that Buttigieg could be the first gay President. I asked him whether he saw that as a measure of how quickly gay identity has become accepted. “Depends where you are,” he said, thoughtfully. “You quickly get plunged into this world where you’re supposed to represent your community,” but at that point he had little experience of the gay community. “Like, I will fight for the trans woman of color, but do I really know anything about her experience because I’m married to a dude?”