The goof that made the world think President Carter had declared war on Arthur Burns and the Federal Reserve Board last week justifies detailed exploration, for it identifies with precision systematic weaknesses at the top of the administration.

It shows a lack of discipline and order, particularly in the critical matter of economic policy. It raises the basic question about Carter that will, in the end, surely determine the fate of his administration.

What looked like the beginning of a public attack began a week ago Wednesday at the daily White House briefing when a reporter asked about the role of the Federal Reserve in raising interest rates. The White House press office, which Jody Powell heads, ralayed the question for comment to Charles Schultze, chairman of the Council of Economic Advisers.

Schultze, who had raised doubts about the Fed's monetary policy in two recent speeches, excerpted from these speeches a formal comment. The comment deplored the recent rise in money rates, suggested that it had a bad effect on the stock market and asserted that, if continued, might adversely affect the whole economy. Those remarks were posted by Powell's office Thursday on a bulletin board in the White House press room.

The posted noctice, coming without warning to a group of reporters unschooled in the background, triggered on Friday a spate of stories to the effect that the White House was publicly chastizing Burns for the Fed's monetary policy. Accompanying the accounts were a number of analytic pieces asserting that the President was getting ready to drop Burns as chairman of the Fed when his term expires on Jan. 31.

Even before the stories broke, Schultze telephoned Burns to explain what had happened and assure him that the administration was not suddenly attacking him. Secretary of the Treasury W. Michael Blumenthal also called Burns to explain. In fact, Blumenthal had more reason for anger than sorrow. Schultze had passed the comment along, and the White House had posted it, without even consulting Treasury.

Over the weekend The Washington Star and The Washington Post carried accounts of what administration officials acknowledged was a "goof." But the semicorrection did not catch up with the original stories, nor did it convince many skeptics who believe that the alacrity of the Schultze comment and the fact that it passed through the Powell office reflect White House hostility to Burns. On Monday, The New York Times carried a lead editorial entitled "Serving Notice on Dr. Burns."

I personally believe there was a miscalculation of press reaction not a deliberate serving of notice. Still, damage has been done by the phony war. The general impression that the administration doesn't know what it's doing, particularly in the economic field, has been reinforced. Secretary Blumenthal has been weakened by renewed evidence that he is not running economic policy. The task, which largely devolves upon him, of winning business confidence, becomes that much harder.

Moreover, it is not as though the "goof" was just a freakish accident. On the contrary, it springs from weaknesses in the way the President manages his affairs.

Carter wants to be boss of economic policy and refuses to delegate authority to a single figure. As a result he overburdens himself with detail, and - since he cannot be master of most of what is going on - opens policy to a free-for-all. Even on such critical matters as reform v. recovery, priorities in economic policy are not clear.

Then there is the causal, off-the-cuff style of the leading White House aides, Hamilton Jordan and Jody Powell. Their style no doubt contrasts appealingly with the Von Moltke-taking-the-orders-of-the-day-from-Bismarck style affeacted by Haldeman and Ehrlichman under President Nixon. But it also breeds a laxity, a lack of seriousness, which lends itself to actions - like the posting of the comment - that are not thought through.

What is required to mend these difficulties is not in doubt. The President ought to designate somebody as his chief adviser on economic policy. He needs a chief of staff who can foster in the President's office a strong sense of discipline. But it is a question - probably the most serious question one can ask about Carter - whether he truly wants strong independent advisers around him.