NEW YORK Presidents, like the rest of us, only learn that their intuitions about the world are wrong when painful experience teaches them so. Some grip their convictions all the harder and blunder onward. Woodrow Wilson could not accept that neither Europe’s leaders nor American politicians shared his vision for a post-war reordering of the world; Lyndon B. Johnson could not admit that he was losing Vietnam; George W. Bush could not see that America’s own allies would not sign on to his swaggering Global War on Terror.

Others adapt, as John F. Kennedy did when his bellicosity almost provoked World War III.

Neither Barack Obama nor his senior advisers are about to admit that experience has forced the administration to reconsider its collective worldview; but it has.

When I interviewed Obama on the campaign trail in 2007, he told me that if and when he became the “face” of U.S. foreign policy and power, people around the world would see that America’s president understood their plight. Although he was careful to add that none of this would matter unless he made “prudent strategic decisions,” Obama believed that his voice and biography would allow the U.S. to recast itself and to climb out of the deep reputational hole into which Bush had plunged it.

Once he had gotten the world to see the United States in a different light, Obama thought, he could make progress on frozen issues like nuclear nonproliferation or peace in the Middle East.

It was this intuition, in turn, that shaped Obama’s response when he was asked in a debate with Hillary Clinton whether he would meet “without preconditions” with the leaders of states like Iran or North Korea. “I would,” he said. This is who he was: A leader who would disdain diplomatic orthodoxy to address states and peoples in a way they had not been addressed before. Thus was born the policy the Obama White House called “engagement.”

In March 2009, Obama sent a New Year’s message to Iran proposing “engagement that is honest and grounded in mutual respect.” Obama continued to hold out hope for a breakthrough, which is one reason he at first declined to criticize sharply the grossly rigged 2009 Iranian elections. Then he spoke out more bluntly, and the Iranians rang down the curtain.

Did Obama really believe that Iran would respond to his deferential blandishments? A senior White House official says that while the administration did, indeed, hope that engagement would work, the “theory of the case” was that “if we can demonstrate that we’re not the problem,” and Iran was, the president would be able to assemble a strong coalition to impose tough sanctions on Tehran. And that is precisely what happened.

Obama’s 2009 speech in Cairo was in many ways the high-water mark of engagement writ large – the transubstantiation of face, voice, and personal narrative into diplomacy. The president offered a “new beginning” to the Muslim world based not on new policies but on a new posture of respectfulness and mutuality. He asked Israelis and Palestinians to engage in a similar act of mutual recognition.

The Cairo speech produced a brief burst of enthusiasm in the Muslim world, followed by disappointment. Negotiations between Israel and the Palestinians struggled at a low ebb and then wheezed to a stop. And no one in the Arab world attributed the great awakening of 2011 to Obama’s modest exhortations.

This was, in effect, Obama’s Wilsonian moment. At the time, Robert Kagan wrote in The Washington Post that Obama seemed to believe, like Wilson, that “nations will act on what they perceive to be the goodwill, good intentions or moral purity of other nations, in particular the United States.” Obama was the incarnation of that goodwill.

In fact, the behavior of states, as Obama discovered, is governed overwhelmingly by national interest. Public opinion does matter, far more than ever before. But what Obama’s Arab public wanted to know was how he would pressure Israel into accepting a two-state solution, or what he would do to dislodge the tyrants who ruled their lives. Obama’s “salaam aleykum” was pocketed, and ignored.

It’s clear, in retrospect, that Obama, like so many American presidents, entered office with too little respect for the world’s intransigence and too much for his own persuasive powers.

But the consequence of Wilson’s, Johnson’s and Bush’s stubbornness or naivete was catastrophic; not so Obama’s. And perhaps because he is a more supple figure than they, he adapted to the world as he found it. By the end of 2010, the White House had announced that it would no longer pursue active talks between Israel and the Palestinians. By 2010 he had gotten all he could on nuclear nonproliferation and let the subject lie. Even when he agreed to join the coalition invading Libya in January 2011, he was the last one on board. Michael O’Hanlon of the Brookings Institution, no devotee of Obama, describes the president as the fastest learner in the Oval Office since George H.W. Bush and, before that, Kennedy.

The Barack Obama of 2012 is preoccupied with ending inherited wars and suppressing terrorism and the threat of Iran – and, of course, the threat of Mitt Romney. Obama’s challenger has done his worst to describe the president as a liberal softie. But Romney is not getting much traction. Obama is still the face of American power – but it’s a very different face.