You may be aware that grisly things happened in the Tower of London in the Olden Days. But do you care? Thousands of visitors pass Traitor's Gate and watch the fat ravens waddle about, look at the simulacrum of Sir Walter Raleigh's room in the Bloody Tower and shudder at the story of the young princes murdered, tradition has it, by Richard III. There's plenty of violent and macabre history here, all of which makes it a great gothic place to visit. But some sensitive souls believe it's time we acknowledged that this ground is sacred to the innocent dead, just like the footprint of the Twin Towers in New York.

At long last, something is being done to correct our shallow assumption that the religious persecutions and marital cruelties of the Tudor age are, nearly half a millennium on, history. If you visit the Tower of London this summer you'll find the site of the scaffold on Tower Green covered by a tarpaulin, and evidence of digging and foundation-making. I was baffled at this disruption, and even more baffled when I read the explanatory notice. But it's all true . . .

On September 4, the Tower will unveil "a permanent memorial to 10 people executed within its walls". It will not be some old-fashioned bronze of Anne Boleyn, but a sophisticated contemporary work by Brian Catling, best known as a performance artist, with an interest in the resonances of London history (he is a collaborator of the writer Iain Sinclair). The monument will consist of "a clear glass pillow resting on two polished discs, one of glass and one of granite". Catling's design was the winner out of five models judged by a panel whose members - including the historian David Starkey and Iwona Blazwick, director of the Whitechapel Art Gallery - considered, among other criteria, how far each design showed "awareness of issues/context".

What? What issues? Don't misunderstand me. The Reformation was a violent era of religious intolerance, Henry VIII obviously a real louse, and the most famous and pitiable of the 10 people to be commemorated - Henry's wives Anne Boleyn and Catherine Howard, and the hapless claimant to the throne, Lady Jane Grey - were victims of bigotry, misogyny and machiavellian politics. But all this happened a very long time ago.

Does that seem an unfeeling thing to say? Memory has become the most sacred and at the same time the most empty value of our culture. It's strange to think that only recently, intellectuals diagnosed western society as a drooling amnesiac stumbling through a shopping mall. "Most young men and women at the century's end grow up in a sort of permanent present lacking any organic relation to the public past of the times they live in," the historian Eric Hobsbawm lamented in 1994. Of course, this is still true, in the sense he meant it. We don't have much critical grasp of history; instead we have replaced it with a cavalcade of collective memory.

The idea that we need to be more contemplative and mournful when we visit a grand guignol tourist attraction such as the Tower of London is manifestly absurd. But our culture increasingly offers no answer to memory, except counter-memory. A radical historian might say it's a disgrace for the handful of royals and nobles treated to private executions in the Tower to be singled out for commemoration when masses of the poor were hanged at Tyburn, on the edge of Hyde Park. How about remembering these working-class deaths? But I don't want to remember them either, thanks. We don't owe a memorial to anyone who died so long ago. They were buried and mourned, their executioners were buried and mourned. It is all over, and nobody alive today bears the least responsibility for anything that happened during Europe's wars of religion.

There's a funny way to look at this, and a sad way. It's comic if you think of it as a parody of the funereal sobriety in which so many artists have indulged ever since Maya Ying Lin's Vietnam Veterans' Memorial in Washington demonstrated how good contemporary art can look in black. Her 1982 wall of names now looks like one of the most influential works of the 20th century. She saw how well mourning becomes minimalism, that pared-down object art of the 1960s whose creators certainly had not seen themselves as inventing a new art of grief, but whose younger revivalists find this mute art can speak silently of death. Rachel Whiteread's early casts, such as Ghost and House, discovered a terrible weight of grief in the most ordinary domestic space. For all we know, the lives that once filled the house she monumentalised in Bow may have been ones of ceaseless joy and laughter, but grief sweated from her cement. She was able to go on from this to create a Holocaust memorial without changing her aesthetic at all.

Such readiness to take on vast historical themes, such determination to dwell on and in darkness, lends art a moral and emotional authority that's hilariously punctured by the Anne Boleyn memorial. The Tower of London is doing us a favour in parodying the mournfulness of Christian Boltanski's Reserve of Dead Swiss and Darren Almond's clock that has something to do with the Holocaust. The Tower's commission makes me wonder if recent art may have a less stylish artistic pedigree than it thinks, for there is already, on permanent view in London, a work that remembers the violence of the Tudor state. It is Paul Delaroche's painting The Execution of Lady Jane Grey in the National Gallery. Most people would agree this is a masterpiece only of 19th-century sentimentality. Could it be that far from fighting the forgetfulness of the late capitalist world, artists who deal in commemoration pander to a neo-Victorian mawkishness?

That's the comic way of looking at it. To see the grim side, watch the news. As we approach the fifth anniversary of September 11 2001, and controversy continues to confound the troubled building site in New York - revisions were agreed this year to Michael Arad's and Peter Walker's design for a monument, Reflecting Absence, to consist of two voids surrounded by trees on the World Trade Centre footprint - it is surely becoming obvious that memory is not always the great humane value we take it to be. Memory cherished too long, with too much bitterness, makes a stone of the heart. It's often better to forget. Art, anyway, can serve us better by criticising memory than by enshrining it in monuments that blindly assert false and vengeful grudges even as they claim to pay simple homage.

Recently I watched Leni Riefenstahl's film Triumph of the Will and realised something that shocked me. I'd seen those terrifying stills from her documentary of the Nuremberg Rally, in which rank on rank of identically saluting stormtroopers form a massed legion of death. But I never knew until I saw the film what they were doing: paying their collective respects at a memorial to German soldiers killed in the first world war.

The problem is not with memory as such but with the misleading notion of "shared" memory. Grief is personal and private. The moment you claim it can be shared by strangers - and this is what all public memorials do - you transform it, you tell some kind of lie.