In a handful of countries applying to join the European Union, the European debt crisis has raised an urgent question: Why would we want to be part of that mess?

The answer offered in places such as Iceland could have profound effects on the health and shape of the E.U.

To understand just how pervasive the debate has become in this small island nation, it’s best to keep walking past the stone Parliament building downtown and forgo the prime minister’s office up the hill. The politicians have settled into their trenches, arguing the issue long enough and loud enough to make any Washingtonian feel at home.

It is in the streets where one realizes how sharply the subject has divided this country of 318,000 people and forced fundamental questions about what sort of future Icelanders want.

“Iceland isolates itself if we don’t participate,” says a 24-year-old bouncer at the English Pub, where a guitar player is belting out covers of Billy Joel and Prince. “The euro is a more stable currency.”

“I don’t want the euro, hell no,” says a 49-year-old woman selling burgers and fries from a food truck in a central square. “The countries that have the euro, it’s going pretty badly.”

Taxi drivers. Tattoo artists. Barbers. Baristas. Nearly everyone has an opinion, and very often a strong one.

“It’s a very hard topic; it’s very divisive,” said Baldur Thorhallsson, a political science professor at the University of Iceland. “It’s about national identity, and that’s why the debate is so heated. It’s a question of where we belong.”

There is an irony hanging over the debate, of course. Three years ago, Iceland’s European neighbors, along with much of the world, looked on in dismay as the island’s once-booming economy imploded, the result of a banking system that had grown far too large and far too reckless.

Now, as Iceland shows glimmers of recovery and Europe finds itself wrestling with a debilitating debt crisis, many Icelanders look at the situation with alarm and disdain. And Iceland is not alone. On Sunday, Croatians voted to join the E.U. next year despite reservations among many citizens about a loss of sovereignty and the debt troubles in the euro zone.

Here, those who favor joining the E.U. — a group that includes Prime Minister Johanna Sigurdardottir — argue that Iceland would benefit from a larger community, primarily by gaining much-needed stability in its currency. Iceland’s krona has long suffered from precipitous booms and busts and has little appeal outside the country, especially after the recent crisis.

Those against joining the E.U. — a group that includes former prime minister Geir Haarde — note that the euro has done little to stave off instability in Europe and argue that allowing the krona to fall in value helped hasten Iceland’s recovery by making its exports cheaper. Many critics also note that Iceland could lose control over the policies that govern fishing off its coasts, a particularly sensitive issue because it involves one of the country’s chief exports.

Ossur Skarphedinsson, Iceland’s foreign minister and a longtime member of Parliament, ticks off a list of reasons he thinks the country should join the E.U. Among them: long-term currency stability, an opportunity to expand the number of industries in Iceland and the potential to attract more foreign investment.

“I am absolutely convinced that they will do whatever it takes to save the euro,” he said of his counterparts in Europe wrestling with the debt crisis. “I am looking further ahead. In the end, Iceland would lose its competitiveness compared to Europe [if it doesn’t join]. We would slowly start to lose the best of every generation. That is my worry.”

But even he acknowledged that, while many Icelanders agree with the need to adopt another currency and have shown enthusiasm for the euro in the past, a vote held today to join the E.U. would fail miserably, possibly by a ratio of 2 to 1.

Of course, any such choice remains well in the future.

Iceland has long had economic agreements that allow it to trade with European partners and allow its citizens to travel, work and study on the continent. The country began its formal application to join the E.U. in 2009, and officials here are negotiating the particulars of a deal with European authorities, a process expected to take at least until next year. New E.U. member states are required to adopt the euro once they meet economic criteria.

Eventually, the question is likely to be put to a vote in a national referendum. The politicians will pontificate. The fishermen and financial firms and unions will lobby. But ultimately, the decision will come down to strong-willed citizens like Margret Sveinbergsdottir and Thora Brynjulfsdottir.

“People are afraid,” said Sveinbergsdottir, 65, sipping coffee recently at the Roma Cafe inside a popular mall just outside the city center. But “our currency would be more stable. We need some more certainty.”

“We want to have our independence,” said Brynjulfsdottir, 50, selling hats at a stand nearby. “We are a small country, and if we go into the European Union, we are going to be lost.”

The women sat in the same corner of the same shopping center on the same rainy afternoon, separated only by a few feet. But like so many citizens here, they remained miles apart.