Several mornings a week, prominent public figures come to the Sheraton-Carlton Hotel to have "Breakfast with Godfrey" -- Godfrey Sperling of The Christian Science Monitor -- and 15 or 20 other reporters. Yesterday morning the breakfast ran 20 minutes over its scheduled length, a most unusual occurrence, but one that seemed justified under the circumstances: for a few moments, it looked as though a politician might pick the Sperling breakfast as the venue for a declaration of candidacy for the presidency.

The politician was Sen. Adlai E. Stevenson of Illinois, a frustrated Democrat who has turned his private agonizing over the course of the Republic into a strange public spectacle during the last fortnight. Stevenson is fed up, and suddenly has decided he has no reason to avoid saying so.

His frustration first erupted two weeks ago in an interview with the St. Louis Post-Dispatch. The entire world, he said then, is "trembling," but America is led by an "embarrassingly weak" president who is "surrounded in the White House by bush leaguers."

"We are dominated by cheap and easy answers, and reorganization plans that tinker with the mechanism of government," Stevenson said, when dire problems facing the country demand "radical answers."

When a senator -- particularly one with a famous name -- talks like that, the Washington press corps lends an ear, then lends encouragement. Stevenson has had a series of interviews in the ensuing two weeks, and then an invitation to the Sperling breakfast. As Stevenson noted there yesterday morning, "Some of my more bizarre statements lately" have attracted unusual attention to him and what he has been saying.

What he is saying is not entirely clear, as Godfrey Sperling himself noted at the end of the breakfast. "It seems to me," the amiable, portly Sperling observed, "that you've dealt too much with nuances...." This was a polite expression of bafflement that seemed to be widely shared around the breakfast table.

Stevenson is unmistakably blunt about his diagnosis of the country's problems, but less precise about how to solve them and about what role he might play in a solution. Running for president, he acknowledged -- conceivably in a third party, but if it's in 1980 then as a Democrat -- was a most tempting possibility.

That is the sort of remark political reporters understand, or think they do. But at breakfast Stevenson quickly muddied the waters: "Everybody seems to assume that the objective [of running for president] is to win," Stevenson complained. Referring to the father whose name he carries -- the unsuccessful Democratic candidate for president in 1952 and 1956 -- Stevenson insisted that winning "wasn't his objective at all."

In his own case, Sen. Stevenson suggested, the objective might be "to find out if an idea can still triumph -- or if it can even make a dent" in the modern American consciousness.

That idea is what remains somewhat nebulous, though Stevenson promises to wrap it into one package for public presentation soon. It is based on the senator's belief that the country is in serious economic peril. He fears that the weakness of the dollar -- which he attributes to America's failure to trade successfully in the world -- has endangered the entire world monetary system. He fears that American technology and science are faltering, and that we have no source of economic vitality now comparable to the postwar technology boom and the '60s space race. "If we lose that [vitality]," Stevenson said in an earlier interview last week, "It seems to me there's not much left."

Stevenson calls for a new trade policy (he will unveil the specifics of that in a week or two, he says), a national oil and gas corporation to deal with many energy problems, and a range of policies to again unleash American ingenuity and competitiveness.

In foreign policy he argues that the presidency has been hobbled by reactions to Vietnam -- reactions "not to history, but to its aberrations." He expressed alarm that the United States cannot react to "Soviet and Cuban imperialism" with both economic power and creative diplomacy.

Concerning the presidency, Stevenson acknowledged that running was "getting more interesting." But he then added that he would hate to contribute to the Republican cause, or even to the cause of other Democratic opponents of President Carter, whom he doesn't approve of either. The Kennedy wing of the party wants to revert to outdated "welfarism," he argues, while others opt for "standpatism."

All of this comes from Stevenson in a low, sometimes mumbling voice. He sat at the breakfast table with the familiar Stevensonian forehead often buried under long fingers. He repeatedly rubbed his eyes, looking forlorn. "I'm not trying to suggest that I am the Ayatollah Khomeini of American politics," he said at one point, mentioning his "weaknesses" as a politician.

He also said he wasn't sure what the country's problem is -- "whether this is a crisis of American leadership or acrisis of the American spirit," as he put it.

As the breakfast meeting went on, some of the reporters present got the idea they might be able to push Stevenson to declare his real intentions. "What you're saying is you won't run for the Senate [his term ends in 1980] but might run for president?" one reporter asked.

Well, yes, but maybe not -- well, he'll have an answer by early April at the latest, Stevenson promised.