I am an admirer of Rudy Giuliani, the greatest mayor New York has ever had, who deserves his place in history whether or not he realizes his presidential ambitions. I am also a fan of the British professor of history at Harvard, Niall Ferguson, whose books — such as “Colossus, Empire” and most recently “The War of the World” — are big, bold, scholarly and rightly popular.

Mr. Ferguson also writes an influential column in London’s Sunday Telegraph. This week he devoted it to a harsh critique of Mr. Giuliani’s “naïve” ideas about foreign policy, and in particular any historical analogies between the present war on terror and past global conflicts.

It is easy for a historian to pour scorn on a politician when the latter ventures into his own field. Mr. Ferguson is worried that one of Mr. Giuliani’s heroes is Winston Churchill. “It shows that Giuliani buys the idea that since 9/11 America has been fighting World War III (or IV, if you like to give the Cold War a number). Time and again in the last six years, leading Republicans have drawn these naïve historical analogies. … The reality is that the threat posed by Islamist terrorism today is wholly different from the threat posed by the Axis powers in 1941-2.”

Now, Rudy Giuliani may not be a historian, but he is certainly not naïve about history. He knows perfectly well that the Islamist threat is different from that of Nazi Germany or Imperial Japan — though there are enough ideological parallels (and in the Nazi case historical links) to make a term such as “Islamofascism” useful. Because this threat is different does not mean it is less deadly.

Though he does not mention it, I cannot believe that Mr. Ferguson is unfamiliar with the work of Norman Podhoretz, whose new book “World War IV: The Long Struggle with Islamofascism” explains precisely why comparisons between the world wars, the Cold War and the present jihad are legitimate. It is no coincidence that Mr. Podhoretz has been advising the Giuliani campaign. Mr. Ferguson, too, admits that he “has been playing a (frankly minor) role as an adviser to John McCain” — a fact that may have something to do with his dismissive tone towards a rival candidate, whose campaign (unlike Senator McCain’s) has gone from strength to strength.

What we have here is really a debate between Messrs. Ferguson and Podhoretz, acting as proxies for their respective candidates — either of whom could be the next president. So what is the Ferguson case against the Podhoretz thesis that the war on terror (a term that not even President Bush finds satisfactory) should really be seen as World War IV?

The arguments advanced by Mr. Ferguson are not very persuasive. It is true that Osama bin Laden claims that he wants to convert rather than conquer the West, but throughout the history of Islam, conquest and conversion have gone hand in hand. Islamofascists would cheerfully annihilate millions of Jews and Christians if they had the means, which they may yet acquire. In some countries, they are already attempting genocide, if not quite as efficiently as their Nazi and Communist predecessors. Just as Stalin, Mao and Pol Pot mainly killed their own defenseless peoples, the Islamofascists kill more Muslims than anyone else.

Mr. Ferguson claims that a total of 6,000 Americans have so far been killed by jihadi terrorists, the Taliban and Iraqi insurgents, compared to 20,000 Allied soldiers and civilians killed on average every day during the World War II. But this does not compare like with like. Americans made up less than one percent of the Allied death toll between 1939 and 1945. Even that toll was far higher than in Vietnam, which yet proved to be unsustainable. Just because relatively small numbers of Americans have died so far does not mean that the war cannot be lost. Indeed, a huge and growing media and academic industry exists to prove that America has already been defeated. Nor is it right to minimize the casualties inflicted by the global jihad. They cannot yet compare in scale or intensity to the carnage of World War II. But they are quite comparable to those of the Cold War, which cost tens of millions of lives over 40 years. It is an ominous sign that the only continent that was spared during the Cold War, North America, was subjected to the most audacious and horrific attack of all. Mr. Ferguson pours scorn on Mr. Giuliani for saying that his nightmare is “an Iranian-made dirty bomb in London or Rome or America.” Well, he is not alone.

And in threatening that, like Ronald Reagan, he would be “a little unpredictable” in dealing with rogue states, Mr. Giuliani is not, as Mr. Ferguson says, historically inaccurate. Reagan’s most memorable Cold War exploits, such as his “evil empire” and “tear down that [Berlin] Wall” speeches, surprised even his own officials. But they worked. Surprising the enemy usually does.

If you want to understand World War IV, read Norman Podhoretz. His brilliant book shows that America is necessarily fighting a different world war with a different doctrine, different means and different aims. What has not changed, though, is the need for courage, resilience, perseverance and all the other martial virtues that made Americans so formidable in the past. As it happens, they are also the outstanding qualities of Rudy Giuliani.