As the Petraeus scandal unfolded last week, we got a crash course on the Tampa social scene, General Allen's superhuman ability to write 20,000 to 30,000 pages of "potentially inappropriate" e-mails, and the romantic attraction of the six-minute mile. What we did not get was a serious discussion of whether it was a good idea to let a warrior-general run the CIA in the first place.

The real Petraeus story is about much more than a seedy tabloid sex scandal. It's about what he did on the job - his brief tenure at Langley and the militarization of intelligence it represents.

To the outside world, intelligence and defense don't seem so different - they're both vaguely national security-ish. But looks are deceiving. Military officers, as Samuel Huntington famously wrote, are professionals in the "management of violence." Intelligence, by contrast, is all about the management of information - how to get it, analyze it, hide it from the wrong people, and share it with the right ones. The Pentagon's primary mission is to fight. The CIA's primary mission is to learn. Fighting and learning are related, but distinct, producing different organizational cultures, activities, and leadership requirements in the Pentagon and the CIA.

Three concerns arise whenever a military leader runs the agency. The first is the risk of tactical tilt - that war-fighter directors will favor tactical military operations over long-term strategic assessments. Even with a $75 billion overall budget, U.S. intelligence agencies cannot do it all: Too much focus on today leaves us vulnerable to nasty surprises tomorrow.

Michael Hayden, another former general who ran the CIA, during the Bush administration, understood this danger precisely because he was an Air Force intelligence officer. Hayden pushed hard to keep the CIA looking at emerging threats, not just today's battles. He knew the most important customer for the CIA wasn't the war-fighter, it was the president. And because he knew it, "the building" at Langley trusted him. But Petraeus was never an intel guy. He was an infantryman who came to Langley from the battlefield and continued to wage war from within the CIA. Under Petraeus, the CIA's paramilitary activities have continued to escalate. The agency routinely conducts and oversees strikes in places like Pakistan, Yemen, and Somalia. Targeted killing is such big business that the CIA is borrowing drones from the military while its own drone fleet is expanding. Many now worry that old-fashioned intelligence collection and analysis in the nation's premier intelligence agency is getting short shrift.

The second concern with a warrior-director is that military leaders can clash with the CIA's culture. The American military prides itself on having a hierarchical, can-do culture. When the boss gives an order, subordinates are expected to follow it, no matter how great the odds of success or how dangerous the circumstance. American forces are the best in the world in large part because of these cherished values.

The CIA has a different cherished value: speaking truth to power. Analysts and collectors are supposed to present information and assessments even if they know the boss won't like it. No one salutes inside Langley. Hierarchy exists, but the culture prizes rigorous debate to sharpen analysis. Intelligence reports have dissenting footnotes. Military orders do not. As Dianne Feinstein, chairwoman of the Senate Intelligence Committee, said of Petraeus before the scandal broke, "I think he's a brilliant man, but he's also a four-star general. Four-stars are saluted, not questioned. He's now running an agency where everything is questioned, whether you're a four-star or a senator. It's a culture change."

The culture change was clearly hard on Petraeus. Determined to win over "the building," Petraeus arrived from Kabul last year without his large coterie of scholar-soldier acolytes. He decorated his CIA office with military mementos. At a recent event, he even weirdly pinned his military medals onto his business suit. The 60-year-old general faced his first-ever civilian job adrift and isolated from everything he had known in a glorious military career.

The bigger question is whether Petraeus was hard on the CIA's culture. It's difficult to know about something so intangible in an agency so secret, but the indicators are not good. As David Cloud and Ken Dilanian of the Los Angeles Times recently noted, word inside the agency was that Petraeus had lived in his deferential military bubble for so long, the former general would "sometimes visibly blanch in meetings when junior officers spoke up to disagree with him." It is also telling that Petraeus didn't sleep with just any woman. He slept with his "biographer," someone he knew would be likely to write hagiography. Broadwell had no writing credentials but plenty of hero worship. That should have raised some red flags as well as eyebrows: A man who selects someone so unqualified to speak the truth of his own life might have difficulty speaking truth to power or rewarding others who do.

The third concern about putting generals at the helm of the CIA has to do with rules. Rules are the lifeblood of an organization, making clear what matters. In the military there are all sorts of rules about appearance and fitness. How fast you can run, how many push-ups you can do, whether your hair is short enough, how neatly your clothes are pressed. Why? Because unit discipline and individual fitness can spell the difference between life and death, success and failure.

Ever seen a bunch of CIA people? Let's just say appearance and fitness do not spring to mind. Instead, CIA rules are fixated on guarding information, everywhere, all the time. Why? Because in intelligence, that's what can spell the difference between life and death, success and failure. It is hard to overemphasize just how seriously security procedures are taken in this world.

Nobody knows yet whether Petraeus played fast and loose with security rules as he was playing fast and loose with his lover. The CIA is concerned enough that it is conducting its own investigation, and the FBI's "closed" investigation of the affair's security implications suddenly is not so closed. The recent box haul from Paula Broadwell's house found classified documents on her home computer. Her clearance has been suspended. Whether Petraeus gave her access to information she had no business knowing remains to be seen.

But here's the thing: Petraeus did not have to give away the nuclear codes or other "vital national security secrets" to have done a serious wrong. In the intelligence universe, any security breach is serious because big secrets can eventually escape through small holes. Anyone who starts thinking the secrecy rules do not apply to them is a ticking security vulnerability, especially if his mistress has become a jilted lover unhinged enough to send creepy/stalky e-mails.

Only time will tell whether Petraeus's indiscretion was marital or more. But it's high time we stopped thinking that generals can always run everything. David Petraeus was a soldier and a patriot. But the CIA was a bridge too far for him.

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Amy Zegart is a fellow at Stanford University's Hoover Institution and the author of "Eyes on Spies: Congress and the United States Intelligence Community."