Syriza, the left-wing party governing Greece, was elected early this year to bring change to a country suffering one of the sharpest peacetime economic declines in modern history. Turns out doing things differently in a currency union that is not also a political union is almost impossible. So there is a fundamental question about democracy in the eurozone. The degree to which it exists is questionable.

None of the amateurish cavorting of Alexis Tsipras, the Greek prime minister, and his crew should be allowed to obscure this troubling fact.

Tsipras is now trumpeting his democratic mandate to negotiate a better deal after Greeks, in a hastily organized referendum, voted overwhelmingly to reject the austerity package that the European Central Bank, the International Monetary Fund and the European Commission insisted was needed to justify plowing further piles of public money into Greece. In essence, Greeks told Germany to get lost.

It is safe to say that Wolfgang Schauble, the German finance minister who, unlike Chancellor Angela Merkel, has essentially had it with Greece, will be unimpressed by this democratic claim. A vote cannot undo a debt or obscure colossal Greek irresponsibility. Greeks were not asked in the referendum whether they wanted to remain in the euro (in which case they would certainly have voted “yes”), but the effect of their 61.3 percent “no” vote is to bring a Grexit much closer.

Should European leaders now allow this to happen — keep the cash spigot from the European Central Bank turned off, watch Greek banks become insolvent in short order, see medicines and imported foods disappear from pharmacies and supermarkets within a week or two, force Greece to start printing i.o.u.s or eventually drachmas that might allow the country over time to devalue its way back to competitiveness? Should Europe gamble that as this scenario unfolds — and Greeks see they were hoodwinked by Tsipras into voting on an austerity proposal when in fact they were voting on whether to keep the euro or not — the majority will rise up and throw out the leftist government for one more amenable to a deal?

Or should creditors, headed by Germany, now cave to Greece — persuaded at last that austerity has its limits and the Greek people have evidently reached theirs, that grievous mistakes have been made by all sides, that the euro may never recover from the loss of one its members, and that, as the International Monetary Fund concluded last week, Greece is almost certainly going to need some debt relief at some stage anyway? Should the troika swallow its pride and say to Tsipras and his ministers that — despite their incompetence, their amateurishness, their arrogance allied to childishness (fatal combo), their insults and their game playing — they have proved their point and won the day and more money is coming?

The decision is not easy. The abrupt resignation on Monday of Yanis Varoufakis, the finance minister, suggests that Greece may now be more serious about negotiation. Much hinges on how expendable Greece, which accounts for just 2 percent of the eurozone’s economic output, is seen to be. In the end currencies are more expendable than countries. Greece will survive without the euro, initially in great misery. The euro may survive without Greece. But, because trust is the foundation of any currency, and joining the euro was an “irrevocable” decision of all its adherents, the euro will have suffered a body blow. It will become little more than a fixed exchange rate system awaiting the next defector.

On balance, Merkel’s concerns about the destabilizing influence of a “Grexit” should prevail over the tough position of her finance minister. The troika should accept the Greek vote, however flawed, and ease the terms of a deal to include some debt relief. This gets back to my initial point about democracy. Europeans want something to give in the austerity that has been the response to the financial crisis since 2008.

Schauble would see such a decision as surrender — and an open invitation to Spaniards, Portuguese and others to vote for populists who will in turn demand concessions from creditors. But it’s the only way to stop Tsipras blaming everything on his favored German scapegoat. If Greece gets a better deal, this incompetent government will have to prove to Greeks it is competent enough to turn the economy around. I doubt that will happen. Tsipras may not survive long.

The alternative — casting Greece to its fate — will see Tsipras turning Greece into Venezuela, railing against the Germans as Hugo Chavez used to rail against the Yankee imperialists responsible for all his country’s woes. Only this will be Venezuela-on-the-Med, with refugees flooding in from the Middle East and North Africa, and President Vladimir Putin doing his worst to suck Greece into his orbit.

It was a sentimental illusion to allow Greece into the euro in the first place, but sometimes terrible decisions have to be managed rather than reversed. This is still such a case.