Special House elections cannot predict the future, but they can influence the present. That’s why what happened in New York’s 26th congressional district Tuesday is a problem for the Republican Party.

Call it a wake-up call or a major setback. Whatever, the victory by Democrat Kathy Hochul in a heavily Republican congressional district, in a race in which Medicare was a major issue, reinforced the reality that the GOP plunged into a debate over entitlements reform without a strategy for winning the battle for public opinion.

Politicians can lead only as far and as fast as public opinion allows. House Republicans, emboldened by their success in the 2010 elections, made the same miscalculation that other politicians in both parties have made, which is to assume a mandate when one doesn’t exist.

The 2010 elections were about many things: high unemployment and economic unrest; government spending and the role of government; debt and deficits; President Obama’s health-care plan; the president’s leadership generally.

Whatever the meaning of the 2010 midterms, Republicans would be stretching to say they had turned their pre-election campaign into a debate over House Budget Committee Chairman Paul Ryan’s (Wis.) plan to transform Medicare. Having failed to do that last fall, they are paying the price now.

Former president George W. Bush made a similar mistake after the 2004 election when he launched his proposal to partially privatize Social Security. He sprang the plan on Congress and the country in early 2005 without having fully aired his intentions during his reelection campaign. His proposal died in the face of solid Democratic opposition and tepid Republican support, particularly in the Senate.

Obama found himself on the wrong side of public opinion with the policies he put in place after his election in 2008. The public wasn’t ready for the amount of spending he pushed through to deal with the recession. Nor was the public enamored with the shape of his health-care plan. The president’s agenda sparked a strong reaction, particularly from the right.

Republican victories last November have changed the terms of debate in Washington. Obama and the Democrats have given ground on spending cuts, far more so than they were willing to do in the past. But while those elections produced a greater sense of urgency to deal with the long-term fiscal problems, little consensus emerged over how to get there.

Ryan earned plaudits for putting a bold proposal on the table to deal with what is genuinely a national problem. But without significant debate and discussion, House Republicans pressed ahead and voted to embrace it, including the Medicare reforms, as it was delivered from the Budget Committee chairman.

Many Republicans not in the House have sought to keep some distance between themselves and the Ryan Medicare plan — offering general support and praise for Ryan’s approach without backing the details. GOP presidential candidates in particular have been wary about tying their futures to Ryan’s particulars and, in coming months, will spell out their own ideas for reforming entitlements.

That’s one reason why former House speaker Newt Gingrich, burned by previous battles over Medicare, said what he said on NBC’s “Meet the Press” two weeks ago, which was that it is risky to impose radical reform without strong support from the public.

Imagine if he were on any of the Sunday shows this weekend and made the same point. The reaction might be far less explosive than it was. For the point he was making — though his inflammatory language compromised his ability to be heard and may continue to do so — sounds even more telling in the wake of Tuesday’s outcome in New York.

Republicans continue to enjoy public support for the broad proposition that government spending needs to be cut to bring deficits under control. The ongoing talks led by Vice President Biden over the terms for raising the debt ceiling will produce more significant reductions in many programs.

But this is a long debate, not one likely to be settled during the debt-ceiling talks. Politically, Obama’s challenge to the Ryan Medicare proposal, and his call to raise tax rates on the wealthiest Americans, has emboldened the Democrats. The Democrats now have some of the same kind of grass-roots energy and focus that tea party activists gave to the GOP during 2010.

Republicans can point to other factors in their loss Tuesday. Among them are the quality of the candidates and the role of a faux tea party candidate who took at least some votes away from Republican Jane Corwin. But that doesn’t negate what happened.

The president of American Crossroads, the conservative group that spent independently in the New York race and that will spend tens of millions in 2012, issued this statement after the results were in Tuesday night:

“The debate over whether Medicare mattered more than a third-party candidate who split the Republican vote is mostly a partisan Rorschach test,” said Steven Law. “What is clear is that this election is a wake-up call for anyone who thinks that 2012 will be just like 2010. It’s going to be a tougher environment, Democrats will be more competitive, and we need to play at the top of our game to win big next year.”

That was wise counsel. Just as the political environment turned quickly after Obama was sworn into office and began to act in 2009, it has turned again in the months since the Republican victories of 2010. Democrats should not read too much into Tuesday’s results. But it is the Republicans who have the most to learn from what happened there.

Republican leaders believe in their agenda and are not likely to back away from it just because they lost one House seat, particularly one that they could very well win back in 2012. But they have not yet won the argument over how best to deal with the country’s fiscal problems. They have accepted the responsibility to propose. Now they will need to learn how to persuade.