Talking to Adm. Mike Mullen, the chairman of the Joint Chiefs of Staff, in his final week in the job, I found myself wondering whether we are entering a “post-military” age, when our top officers understand that the biggest problems can’t be solved with military power.

Time and again, versions of this theme surfaced in my conversation with Mullen. He has been, by widespread assessment, a very effective chairman who restored the position to prominence in national security decision-making. But the problems he’s leaving unresolved lie at the periphery of the military space, where conventional weapons can’t reach.

Time magazine’s Joe Klein once likened Mullen to a “country doctor,” and that’s what he has been for the military — a big guy with a doughy face and syntax that doesn’t always parse, but who looked like the commander and didn’t bend, more than was appropriate in our system, to politicians.

Military officers are by nature problem-solvers who like to fix things, or shoot them, or get around them some other way. So what brings a smile to Mullen’s face, right off, is the feat of sheer military prowess in the May 2 raid that killed Osama bin Laden. Mullen remembers the mess that was Desert One in 1980 — helicopters that didn’t work, aircraft that crashed, shortage of parts, bad training. Watching that fiasco, he recalls, “I sensed that we were in trouble as a military.”

In the Abbottabad raid, it was obvious that this problem of competence is largely fixed. Every night, U.S. Special Operations forces conduct missions almost as complicated as the bin Laden assault. Mullen describes today’s military machine as fearsomely efficient: “a combat-hardened, combat-experienced, extraordinarily professional, competent, all-volunteer force.”

But what are the deeper, intractable problems facing Mullen’s generation of officers? They are about culture, and governance, and the subtle psychological factors that keep people from doing what’s in their interest.

The biggest frustration of Mullen’s four years as chairman was surely Pakistan. He decided early on to forge a personal relationship with Gen. Ashfaq Kayani, the army chief, in the hope that it might be a solid bridge between the two countries. Mullen says that he didn’t realize the extent of the “trust deficit.”

Increasingly, it became clear to Mullen that Pakistan’s problems were embedded in the economic, political and cultural fabric of the country. They’re on “a declining glide slope,” Mullen explains, and this isn’t something America can fix.

Yet hope springs eternal in the military heart. I ask Mullen if Pakistan “blew it,” but the admiral insists the story isn’t over: Kayani still wants to cooperate; Gen. James Mattis, the head of U.S. Central Command, just had a good visit there last weekend; and so on.

And then there’s Afghanistan. Mullen insists that this isn’t just an expensive stalemate, that “the trends are good” and “it has moved in the right direction.” But he knows, too, that the definition of success is to transfer responsibility to an Afghan government enfeebled by problems of governance and corruption. In that sense, all the brilliance of the American military won’t be enough — not when the definition of victory is so interwoven with politics and culture.

What troubles Mullen is that this magnificent professional force has become a separate tribe in America, too little connected to the rest of the country: “They don’t know the depth and the breadth of what we have been through, the numbers of deployments, the stress on the force, the suicide issues, the extraordinary performance.”

Mullen knows that his greatest legacy will be a cultural and legal issue — ending discrimination against gays in the military by dismantling the “don’t ask, don’t tell” policy. He did it for reasons of conscience and never looked back. It was a moment of leadership, pure and simple.

As Mullen prepares to leave Friday, the federal government is shuddering with the politics of paralysis. So I ask him, as a last question, about the political divisions he has tried to bridge as a nonpartisan chairman. He muses that it’s odd to be lecturing Iraqi Prime Minister Nouri al-Maliki about governance when “there’s a lot of things we don’t get right” at home.

What America needs, he says finally, is the same requirement that makes the military work, which is “accountability for outcomes.” A political system that works — whether it’s in Islamabad or Kabul or Washington — is one that takes responsibility for solving the problems that do not yield to force of arms.