At one end of the US war machine are people like Donald Rumsfeld, the ultimate defense intellectual who views the war on terrorism as an intriguing puzzle requiring new ways of thinking. At the other are the long-serving men in uniform such as General Tommy Franks, the former artillery officer leading the campaign.
Gen Franks is the commander-in-chief of the central command, whose headquarters are in Tampa Florida, from where he is orchestrating the air strikes on Afghanistan. He is a blunt, outspoken veteran of the Vietnam and Gulf wars and, by all accounts, he has taken to heart the lessons of both: be very sure of what you are doing before you put soldiers on the ground, and rely as much as possible on the awesome destructive capability of US air power.
The two men embody the different approaches circulating in the corridors of the Pentagon over how to pursue the war on terrorism. Winter is coming to the Afghan highlands and decisions have to be made quickly, but a week's bombing under Gen Franks's command has so far failed to push Osama bin Laden or the Taliban leader, Mullah Mohammed Omar, into the open where they could be picked out by an air strike, or grabbed by special forces.
That would have been considered a bonus in the initial phase of the campaign, but in the absence of such a stroke of luck, differences over how the plan should proceed have come to the surface.
Mr Rumsfeld and his civilian advisers believe the US military does not have the flexibility to combat an enemy like Bin Laden. They point to a computerized war game in 1997 in which the army took on a terrorist organization similar to al-Qaida, and lost. The generals, the analysts concluded, spent too much time looking for things to bomb, and not enough time looking for innovative methods of eliminating the enemy.
Mr Rumsfeld is reported to be so frustrated with the pursuit of the war by Gen Franks's command, with its emphasis on waves of Gulf-style bombing sorties, that he is pressing to have operational control shifted from Tampa to Washington. Mr Rumsfeld and his circle want to pursue a new military doctrine built around small groups of special forces soldiers who will dart in and out of Afghanistan looking for intelligence and targets.
Uniformed top brass are more comfortable with the technique of the Powell doctrine - named after secretary of state, Colin Powell - which dictates the overwhelming use of air power until the deployment of ground troops is either unnecessary or met with minimal resistance.
This week US and British special forces units are expected to be deployed in Afghanistan, but they are being sent on highly dangerous fishing expeditions, concealing themselves along the sides of dirt roads and mountain paths on the chance that Bin Laden or Mullah Omar, or their top lieutenants, might pass by.
Senior Pentagon officers have pointed out the dangers in such missions. The terrain is littered with millions of landmines, and "butterfly" anti-personnel mines, dropped by Soviet helicopter pilots over hostile territory in the 80s.
Before sending in larger numbers of troops, the traditionalist generals want to continue the air campaign. It has been kept up for seven days, with only a pause on Friday, the Muslim day of prayer.
But such niceties are not helping the state department efforts to keep the international coalition together.
At the weekend the Pentagon admitted that an F-18 navy strike aircraft had accidentally dropped a 900kg (2,000lb) bomb on a suburb of Kabul, killing four civilians and wounding eight. Latitude and longitude were mixed up when the coordinates were entered into its guidance system.
The Taliban are claiming that civilian victims have been more numerous. In any case the distinction between combatants and non-combatants is blurred. Many of the "troop concentrations" targeted are conscripts who may have been market vendors only a few days earlier and who were rounded up by Taliban press gangs.
These troops have been hit by cluster bombs and on one occasion by a huge bunker-buster bomb which would have burrowed into the ground beneath them and then swallowed them as the explosion opened up a gaping crater.
As reports of the casualties percolate into the Middle East and Pakistan, support for the US is fast eroding. A poll of Pakistanis found that 83% supported the Taliban in its confrontation with the US. According to Newsweek, which conducted the poll, support for the Afghan militia jumped by 40% when the bombing began last week.
The Taliban are beginning to exploit the TV images of US mistakes by inviting reporters to view the damage. This "collateral damage" is inevitable in a bombing campaign. The only way to avoid it is to put troops on the ground, but that is fraught with human, military and political problems. The US population remains virtually unanimous in support of the campaign, but that may change with the return of body bags.
The Pentagon's military leaders have painful memories of the last two comparable special forces missions, which both ended in fiascos - the 1980 "Desert One" operation to rescue US hostages in Iran, and the 1993 raid on Mogadishu, Somalia, by Rangers and Delta Force commandos, which failed at the cost of 18 dead, 73 wounded, and two helicopters shot down.
Some in the Pentagon believe Bin Laden may not be in the caves of the Hindu Kush after all, but could be hiding in the warren of slums outside Kandahar. There, he would probably be protected by fervently committed guerrillas. Going in after him would be an operation reminiscent of the Somalia disaster.
© Guardian Newspapers Limited 2001