Perhaps decades too late, European leaders are waking up to that realization today.
Mass migrations of Muslims to Western Europe over the years have raised the threat of "homegrown" terrorism in European capitals and created "ghettoes" of isolated and sometimes hostile enclaves that, in some cases, have taken over entire sections of cities.
It's not a healthy situation for the immigrants, as they remain poor, detached and unassimilated. Nor is it healthy for the particular nations -- as they play host to increasingly angry and often costly multitudes who add nothing of substance to the larger community.
The sad irony of multiculturalism run amok is that it endangers cultures: The nations of Western Europe are threatened with having their own cultures and values diluted or worse.
The results are obvious and troubling. Muslims in England burning the nation's flag, eschewing the long and storied traditions and laws of Britain, and the host nation embarrassed by itself: the very flying of the Union Jack has become controversial in some quarters, and Scotland Yard even banned officers from wearing it in 2009, until the policy became widely known.
Belatedly, Europeans are beginning to fight for their own cultures.
In east London, residents have fought a land-use permit for a mosque because it doesn't allow women to worship. And in speeches from England's David Cameron and Germany's Angela Merkel, lines are being drawn in the sand.
One report says mass migrations have led to "a growing minority that lacks basic language skills, isn't able to find a job and overburdens national welfare systems."
Merkel told a German audience last October that multiculturalism there "has failed, utterly failed." She got a standing ovation. A news report noted a study that said "more than 30 percent of people believed Germany was 'overrun by foreigners' who had come to Germany chiefly for its social benefits."
In his recent warnings about the excesses of multiculturalism, Cameron argued that immigrants should learn the language of the host country and immerse themselves in "elements of a common culture and curriculum."
Cameron urged his large Muslim population not to tolerate Islamic extremism in its midst, and said greater Britain must "confront, and not consort with" even " the nonviolent Muslim groups that are ambiguous about British values such as equality between sexes, democracy and integration."
"He wants his country to be a melting pot and not descend further into a mosaic," as one newspaper put it.
He's absolutely right. Our wildly varying cultures should be celebrated; most of them are beautiful, and the world's diversity is one of life's richest flavors. But what a tragedy if we allowed rampant immigration to make us all the same.
"The truth," French President Nicolas Sarkozy said recently, "is that in all our democracies we have been too preoccupied with the identity of those who arrived and not enough with the identity of the country that welcomed them."
Do we really want to live in a world in which being French or English or German is only a label on a passport?