So who are we?

It's a question simply too difficult to answer, says Tom Lubbock, as a fascinating new exhibition opens exploring the nature of identity.

I went into the passport office to get a passport and there it was, pinned up on the wall: a poster advertising the coming identity card. I haven't seen it anywhere else. I think it may have been a try out. But what it said was: "The identity card. Celebrating individuality. Protecting community." Seriously, that's what it said, with those New Labour verbless present participles. The slogan was illustrated with a chequerboard of contemporary British faces, diversity itself, to show you what a community of individuals looks like. But I ask you: celebrating individuality? An identity card? What kind of mug-shots do they think we are?

Identity: it's a word that means all sorts of different things, some nasty and some nice. There is identity as in ID parade. There is identity as in Gay Pride March. The Home Office ad was hoping to give the identity card the cheerful connotations of a Gay Pride badge, something like that. But the word simply is slippery, and so is the concept. The new thematic exhibition at the Wellcome Gallery, Identity: Eight Rooms, Nine Lives, makes the most of this slipperiness.

There are eight separate wooden structures (the rooms) each with an entrance. Go inside and you'll find a miscellany of exhibits accompanied by some talkative explanatory captions. Each room, it emerges, is devoted to an identity-related topic. There's diaries. There's stage-acting. There's brain reading, from phrenology to MRI scans. There's eugenics. There's DNA profiling. There's transsexuality. There's twins.

You also notice that at each entrance there's a portrait, a bust, painting or photo. Each topic is headlined with a key individual (the lives). Pepys stands for diaries, Francis Galton for eugenics, April Ashley for sex changes, Fiona Shaw for acting. Twinship is represented by Charlotte and Emily Hinch (the eighth and ninth lives). But are they really twins? Through a piece of IVF legerdemain, Charlotte and Emily, though conceived from the same batch of fertilised eggs, were born three years apart.

It's a widish range, but of course it could be vastly wider. Identity: the theme teems. Take a minute, and you could cook up another show, equally relevant, completely different. It could have names. It could have personality types. (Are you an extrovert, cerebrotonic, choleric Taurean?) It could have brainwashing and reincarnation. It could have children raised by wolves. And so on. Almost anything you can think of could be connected somehow to our identity.

"Identity" is like a Time-Life book made real. It isn't exactly fun and it isn't exactly serious. It's interesting. It's got a lot of thought-provoking stuff, pointing vaguely towards some big questions, before rapidly moving on. What is the self? Can it be measured? Can it be changed? Who am I really? The exhibition is punctuated with mirrors, some ancient and tarnished to blackness, some new and fancy. Who can resist the "true mirror" that reflects your face the right way round?

But the relationship between verbal and visual here is never happy. The exhibits don't make sense without the captions. When you start reading them, they never say enough. You don't acquire real knowledge. You get thought-bites, factoids, a little learning. It's one of those shows that you come out from with a lot of stories, each one beginning "apparently... "

Apparently, they've done scans that prove that Westerners' brains are more individualistic than Chinese brains. Really? Apparently. And apparently, Francis Galton – he was the Victorian polymath one who coined "eugenics", he was into inheritance and race, he virtually invented fingerprinting, and he went about measuring everything – apparently Galton took measurement so far that he devised a way of quantifying the incremental boredom amongst the audiences attending Royal Geographical Society lectures. Apparently.

Galton's career is full of good apparentlys. He designed a gadget to measure the grip of a handshake. He made composite photographs, where the faces of scientists or murderers were superimposed, to reveal (he supposed) the essential features of these groups. He invented high-pitched whistles to test human and animal hearing – still in use.

Galton is the villain of the show. He believed in fixed selves and types, born geniuses and born criminals, and making sure that the sub-human didn't get born. "Identity" prefers a more fluid picture. There's DNA showing how human genes have roamed all over the world, scattering any idea of racial purity. There's Claude Cahun – another life – the lesbian Channel Island surrealist who played with masks and professed her gender as neutral.

But there are times when the issues and topics let up, and the exhibits stop being pieces of evidence, and become simply a thing to astonish. A video shows Fiona Shaw performing part of The Waste Land – the two sequences starting "My nerves are bad... " and "When Lil's husband got demobbed... " They're passages full of different voices, and she throws herself into all of them, and gears through the transitions between them too. It's intelligent and it's like diabolical possession. If you have to act it – it's probably better not to – this is the way to do it.

Or, in the room of April Ashley, the most absorbing exhibit is her appearance on the The Russell Harty Show (TV chat-show of yore). The pioneering transsexual herself is charming. But for a real study in identity, look at Harty. Here's a non-out gay man, prominently in the public eye, working in that dangerous historical interval when you couldn't be out but you could be easily outed (as Harty eventually was), whose job is to probe others, and whose trick was to be pretty camp. It's a high wire act.

Or go into diaries. Nowadays everything can be confessed, all the time. But if you can drag yourself away from the frighteningly self-revealing occupants of the Big Brother diary room, pay a moment's attention to the experience of Clive Wearing. A moment is all he ever knows.

A musician, who worked for the BBC, Wearing got a virus that struck his brain, giving him a rare form of amnesia. His handwritten diary is one of the most extraordinary bits of text I've seen for quite a while. Here is part of the entry for 21 April 1990:

"11.06 am. Now I am nearly completely awake.

11.30 am. I return just as I am almost perfectly awake.

11.34 am. First, almost totally conscious stroll. I am perfectly awake.

12.39 pm. Now I am almost really perfectly awake.

1.20 pm. Now I am almost actually completely awake."

And it's weird that all these nearlys and almosts are inserted into the sentences like qualifying afterthoughts. But so he goes on.

"3.26 pm. Now I am nearly perfectly awake.

3.51 pm. Now I am nearly completely awake.

4.09 pm. Now I am nearly totally awake.

4.20 pm. Now I am nearly overwhelmingly awake.

5.19 pm. Now I am nearly superlatively awake... "

Next time someone tells you that we should all try to live in the present, you'll know what they mean.

Identity: Eight Rooms, Nine Lives is at the Wellcome Gallery, London NW1 (020 7611 2222) to 6 April, free admission