When Barack Obama delivered the speech to the Democratic party convention in 2004 that launched his national career, he began by telling his own compelling personal story: "Let's face it, my presence on this stage is pretty unlikely. My father was a foreign student, born and raised in a small village in Kenya … While studying here, my father met my mother. She was born in a town on the other side of the world, in Kansas."

He went on to tell his parents' story as a quintessentially American tale of love, hope and aspiration. "My parents shared not only an improbable love," Obama told the rapt Democrats, "they shared an abiding faith in the possibilities of this nation. They would give me an African name, Barack, or 'blessed', believing that in a tolerant America, your name is no barrier to success."

But as Obama prepares to celebrate his birthday on Tuesday there are others who would suggest that his appearance that night was not just probable but plotted – part of a long-running conspiracy by foreign Muslim forces to take over the United States. In the alternative version, his white grandparents were so displeased at the race of his mother's fiance that his mother fled to Kenya. Once there, she was repulsed by the manner in which Muslim men treated their wives, but was now too pregnant to fly home and so remained in Mombasa, where the heat brought on early labour. A local imam was kind enough to lead the festivities and called the boy Obama.

The details change, but the basic storyline remains the same. Obama should not be president because his occupancy of the White House contravenes article two, section one of the US constitution, which stipulates that only a "natural-born citizen" is eligible for the presidency. He is not just un-American but non-American; a faux candidate foisted on America by way of Mombasa rather than Manchuria. Such are the claims of the American "birther" movement.

The aim here is not prove these people wrong. That has been achieved several times over. For them to be right, Obama would have had to persuade the state of Hawaii to collude in forging a birth certificate that has been verified by its Republican governor and director of health as well as the nonpartisan factcheck.org. Moreover, his mother would have had to have the foresight to place birth announcements claiming he was born in the US in both the Honolulu Advertiser and the Hawaii Star Bulletin, 48 years ago, in anticipation of a future presidential run – otherwise, why bother? When you think of the time and effort that must have gone into this cover-up, Obama's election must go down as the most elaborate affirmative-action sting in US history.

Facts won't budge them. The smart ones insist they are just doing him a favour. "What I don't understand is why he hasn't produced [his birth certificate] to get this noise out of the way." Cutting "legitimate doubt" from whole cloth, they create accusations to which the only defence would be to disprove a negative. (How do we know Obama was not enrolled into a school in Indonesia as Barry Soetoro?) Posing as reasonable people asking reasonable questions, many insist that their interest is not in stoking the controversy, but ending it. "I do believe the president is a citizen of the United States folks, don't you?" asks CNN's resident xenophobe, Lou Dobbs. "But I do have a couple of little questions, like you. Why not just provide a copy of the birth certificate?"

When proof is provided, the inconvenient evidence is denied, parsed, undermined or overlooked. Hawaii has produced a certificate of live birth which it both issues and accepts as proof of citizenship. So the birthers demand his full certificate and claim that the document provided is a fraud. Meanwhile, in the absence of tangible proof, birthers are sustained by claims that are variably random, unsubstantiable, insubstantial, untraceable or incredible – and often all five. In November the grandson of the Kenyan imam who allegedly delivered Obama in Mombasa was reported to be on his way to England to claim asylum because he feared the Kenyan authorities would silence him. Which brings us on to the final, crucial part of the birther identity: victimhood. The leftwing media are hounding them and the government is marginalising them. If you can't say what happened to the grandson of that Kenyan imam you'd never heard of, then how do you know the authorities didn't finish him off? And so they turn banality into controversy, truth into speculation, certainty in doubt and the world on its head. Having made up the news, they demand to know why no one is reporting it.

So why dignify these people with column inches when you could just laugh and move on? If they truly are brain-dead, then surely the oxygen of publicity only keeps their contributions in their present vegetative state. There is something to that. But while to engage them is clearly futile, to dismiss them would be reckless, for two main reasons.

First, the birthers are anything but a fringe group. They have found a sizeable audience for their fantasy. A poll last week showed that more than half of Republicans either believe Obama was not born in the US (28%) or are not sure (30%). Mainstream anchors on CNN and Fox routinely give them credibility. So far, 11 Republican congressmen have signed a "birther bill" that would demand a birth certificate from all future presidents. They may have no more credibility than the 9/11 truthers or those who denied the moon landings, but they certainly have more reach.

Second, however marginal they appear now, they were effectively running the country between 2000 and 2008. It was their birther logic (an oxymoron if ever there was one) that provided the mindset, legwork and frontline troops for the Bush era. Iraq was invaded because it could not prove that it did not have something it truly did not have. "We would say, 'Iraq should present any anthrax'," explained UN weapons inspector Hans Blix shortly after the invasion. "While the US and UK were inclined to say, 'Iraq should present the anthrax.'" Guantánamo Bay is still full of people who were incarcerated because they were not able to prove they were not guilty and whose guilt was to some extent inferred by their incarceration.

The birthers' claims might be crazy. But so was Whitewater, which ended with Clinton's impeachment, and the Swift Boat saga helped torpedo John Kerry's presidential campaign.

A senior Bush aide once ridiculed a New York Times reporter over his adherence to "the reality-based community", which he described as people who "believe that solutions emerge from your judicious study of discernible reality". "That's not the way the world really works any more. We're an empire now, and when we act, we create our own reality. And while you're studying that reality – judiciously, as you will – we'll act again, creating other, new realities, which you can study too, and that's how things will sort out. We're history's actors … and you, all of you, will be left to just study what we do."

This is what they do. Even a brief study would show it is no laughing matter.