What does the British Museum have to do with Britain? Not much, some have argued, looking at the worldwide scope of the collections and the seemingly small role played by British things and British history within it.

True, the British Museum is not, and never has been, a museum whose central theme is 'our island story' in that now rather quaint sense that so appealed to late 19th- and early-20th century taste at a time when, though the country was at its imperial zenith, British middle-brow historical and cultural horizons were as insular as never before.

Look at the connotations of 'British' before and after that moment and we find very different sets of resonances. When founded by an Act of Parliament in 1753, the title 'British Museum' was understood in a civic, not national, sense. It was the British Museum because it was not the royal museum.

It was to be a collection for a new country, whose constitutional unification had only recently been tried in the civil war of 1745, when the issue at stake was the critical choice between parliamentary sovereignty under the Hanoverians and Stuart dynastic absolutism.

A peculiarly British form of progressive Protestant parliamentarism won and, in proclamation of a new civic culture for the newly reunited kingdom, eight years later it founded the British Museum for all its citizens.

And, as this new culture was being formed in the intellectual crucible of the Enlightenment, the focus of the collection was decidedly not on the histories, narrative or natural, of Britain alone, but on the whole world within which Britain was playing an increasingly important part.

Already in 1753, London was the largest city in the western world. By 1800 its population would reach a million, a large proportion of whom were migrants from elsewhere in Britain and abroad - Scotland, Ireland, eastern Europe, North America, Africa and the Caribbean.

Britain was shaping the world, but the world was already starting to shape what it meant to be British, something that was made possible by the fact that the roots of Britishness itself were civic, not ethnic.

In 2006 London, and through it Britain and Britishness, are again being transformed, or rather are being brought back to their starting-point, by migrants from these same places and many more besides. Memories are short and perspectives abridged. We habitually contrast the diversity of our present day with an imagined cultural homogeneity loosely situated in the early 1950s, when Britain was supposedly populated solely by white Anglo-Saxons.

This is a myth. The arrival of the SS Empire Windrush in 1948 marked not the beginning of a new story, but the resumption of an old one that had already begun in the late 18th century, the story of the peopling of the world by Britain, and of Britain by the world. This is the national narrative to which the British Museum, and many others throughout the country, was founded to bear witness.

If no longer the largest, London is still the world's most diverse city, ethnically and linguistically. The international engagements of a museum such as the British Museum with partners in Africa, China or the Middle East, naturally include within their scope the London and UK-wide communities from these regions, not just because we believe in community outreach, but because they are the human counterpart of the collection here in Britain.

From Ngati Ranana London Maori Club to the Iran Heritage Foundation, the worldwide origins of London's people and of the museum's collection testify to the historical beginnings, and current reality, of Britishness as an extrovert, composite identity based on a set of civic values accessible to all.

The unexpected recurrence of an expansive, Enlightenment sense of Britishness was of course central to the recent success of the London bid for the 2012 Olympics. Here as nowhere else the games could be celebrated in a city whose existing population convincingly represents the whole world in one place.

Museums can, and should, respond, in thinking both about how they are governed and what they do. The British Museum claims to be a museum of, and for, the world, and trustees of such a collection need to reflect that. Nine of the British Museum's current board of 23 have their origins outside Britain, which is one of the features of the group that makes it so characteristically British.

Edmée Leventis is the Cypriote ambassador to Unesco; Philip Lader is the former US ambassador to Britain and is joined by the American-born author and critic Bonnie Greer; Chief Emeka Anyaoku is a former commonwealth secretary-general; Vikram Seth is the Indian-born author whose work serenely transcends national categories; Ronald Cohen was born in Egypt; Hasan Askari in Pakistan, Olga Kennard is from Hungary; and Lord Broers was born in Calcutta and educated in Australia.

This is a group whose diverse origins mirror the collection which they hold in trust for the world.

Lending objects from the collection has also taken on a new significance as one of the main things we are here to do in order to maximise access to the collection for worldwide public benefit. On average, the British Museum lends 3,000 objects a year to 300 different venues. At the time of writing, loans are underway to institutions in five continents.

At the end of this month, the exhibition, Hazina: Traditions, Trade and Transitions in East Africa, will open in Nairobi, Kenya. Curated and conceived by colleagues from the National Museums of Kenya, it consists of objects selected by them from the British Museum to tell a story for Kenyan audiences about the East African context of Kenya's cultures.

This, we expect, will provide a new model for partnerships elsewhere in the world as the collection fulfils its purpose to be a library of the world's material cultures, available for the world's citizens to consult and make of them what they will.

Which was, after all, the founding vision of this place and, in many ways, of the British museum tradition as a whole. Again, the 'British' in 'British Museum' carried, and may still carry, an open and ethical sense, not a closed, national sense.

Hans Sloane's principal desire, in seeking to maintain the integrity of his collection after his death, was that it should be maintained for the 'improvement, knowledge, and information of all persons' which, in the words of Gowin Knight, the museum's first director, meant 'as well natives as foreign'.

Here, if anywhere, is an expression for museums of the best of what 'British' can mean, in which all may participate.

Neil MacGregor is the director of the British Museum