When you approach the desk at the Pitt Rivers, the question they are expecting is, "Where are the shrunken heads?" The heads are the star exhibits—they inspired a scene in Harry Potter!—but the museum is sensitive about them. When we wanted to take pictures for this article, they said, "Of course, anything you like, except the heads." After all, they are human remains. They deserve respect. So, with respect, we have no photographs, only my description.

The ground floor of the Pitt Rivers is an enchanting, crepuscular clutter. The display cases have titles like Puppets, Instruments of Divination, Lamellophones (surprisingly numerous and widespread) and—the one you’re looking for—Treatment of Dead Enemies. It could be an aquarium designed by Hieronymus Bosch. A dozen shrunken heads float fishily from wires. One has a mop of black hair like a bio-engineered Beatles souvenir, another a carrying handle thrust through its nose, like a fashion accessory from Mordor. Their faces are full of character. There’s no doubt that family and friends would recognise them.

Most of the heads were made by the peoples of Ecuador and Peru. Until the 1960s Shuar men fought a running war with the Achuar, taking heads and turning them into tsantsas. These aren’t really trophies. Making tsantsas was a kind of post-mortem adoption process whereby the soul of a dead enemy becomes part of his conqueror’s family. Once a tsantsa was complete, the shrunken head had no value—it had done its job. So Shuar and Achuar warriors were quite happy to trade them, which is how they ended up here. The heads were so popular that people took to making and trading fakes. The one donated by Pitt-Rivers himself is made of a sloth's head. Others were made from the heads of poor people stolen from mortuaries. We know this because they are stuffed, not with rainforest vegetation, but with a copy of the Quito Times.

The presence of the heads raises questions about what a great ethnographic collection is really for. Does this cabinet of day-to-day wonders help us to understand other cultures better, to uncover our common humanity? Or does it titillate our sense of superiority to the exotic "others" who made them? It was definitely the promise of the exotic that first brought me here. I was a student in Oxford in the early 1980s. When I wanted to impress a girl, I offered to take her to see the shrunken heads. Although I would then demonstrate my more sensitive side by pointing out the interlocking chi balls—11 filigree ivory spheres, delicate as lace, one inside the other, all carved from a single piece of ivory. A miracle. No wonder she married me. But then the whole of Oxford seemed exotic to me then. I spent my first year in a kind of belligerent daze. I couldn’t believe that (a) it was all so beautiful and (b) I was allowed to wander round it. I sought out arcane routes across town, crossing obscure quadrangles, cutting through hidden cloisters, going to libraries and faculty buildings not my own, pushing deeper into the labyrinth, and pushing my luck—waiting for someone to challenge my right to be there. That's how I found my way to the little door at the back of the Natural History museum that leads to the Pitt Rivers.

Nowadays it's well signposted and staffed. Back then it took a bit of nerve to step out of the soaring daylight of the Natural History museum into the shadows of the Pitt Rivers. A museum hidden tucked inside a museum. Inside there were more doors to dare. The cabinets had drawers—was I allowed to open them? In the corner was a glass case covered with a curtain. I pulled the curtain aside and found a vast red and yellow cloak, an ‘ahu‘ula made in the 1830s for Queen Kekauluoki of Lahina in Hawaii, of hundreds of thousands of tiny red and yellow honeysucker feathers. These were supplied by specialists who plucked a few from each bird before letting it go. Each feather is tied into place. The priceless finished ‘ahu‘ula was one of the last to be made. The species of honeysucker from which the feathers were taken is now extinct.

Like an ivory ball inside another ivory ball, each exhibit is wrapped inside a sequence of stories. First the story of how it was made. There's the policeman's amulet made from an old coin and a piece of rope from the neck of "Campi"—a robber hanged for murder in Paris in 1883. Having been shriven for his execution, Campi was in a state of grace, so his relics might have curative powers. The great Haida totem pole was commissioned by Chief Aniithlas to celebrate the adoption of a daughter. I think of the little girl standing outside the chief’s "Star House" with the lowering history of her new family staring down at her through the eyes of cedarwood ravens, bears and beavers.

Then there are the stories of how these things got here. Sometimes the labels give poignant hints. The Fijian necklace of sperm-whale teeth, given to the Reverend Calvert around 1874, was donated years later by his granddaughter in memory of her son, Pilot Officer James Lionel Calvert, who died on active service in 1939. The beautiful Benin plaques were acquired as a result of the punitive British expedition of 1897 in which Benin City was razed to the ground, its art treasures stolen and auctioned off to offset the costs incurred by the destruction.

Twenty thousand of the objects were donated by Pitt-Rivers himself. He was born Augustus Lane-Fox, in 1827, but changed his name in accordance with the terms of an unexpected and colossal inheritance that transformed him from an officer in the Grenadiers into a collector on an historic scale. Hidden in the heart of every museum is one invisible but crucial exhibit—the original Idea which brought the objects together. These Ideas are often more strange and antiquated than the objects themselves. Pitt-Rivers's Idea was unusually complex. Despite the plaques and the shrunken heads, he was drawn not to plunder and treasure but to the mundane. Most collections in the 19th century told stories of race and nationality—Egyptians and Greeks, Romans and Vikings etc. Pitt-Rivers organised his thematically—money, weapons, saddles, hunting and so on. Those shrunken heads sit alongside the heart of an Irish warrior that was (literally) salted away inside a heart-shaped lead casket, a helmet from the Great War and images of Cromwell's head on a pike. European tarot cards share a space with Congolese divination bones, Yorkshire funeral biscuits with Chinese burial headdresses.

Pitt-Rivers was no early cultural relativist. He was an enthusiastic Darwinian. Certain over-enthusiastic Christians get too literal; over-enthusiastic Darwinians tend to be too metaphorical—taking the literal truth of evolution and applying it to culture. Pitt-Rivers arranged his objects into "types". He proposed a science of "typology" that showed how day-to-day objects evolved like species. Of course they evolved faster in some places than others. "Typology", he wrote, "forms a tree of progress, and distinguishes the leading shoots from the minor branches. The problems of the naturalist and those of the typologist are analogous."

He was less concerned with proving the superiority of the West than with proving that real progress was slow and thus violent political change was against the laws of nature. His typological sequences are sentences pleading for reform over revolution.

Today these sentences are inaudible, and the collection sings a different song. The overwhelming impression it leaves is that no matter where we fetch up—on a shingle strand in the Arctic, or a patch of rainforest—we are driven to create. We need to make things beautiful almost as much as we need to make them edible. There are amazing coats made of seal-intestine from remotest Alaska. There are rattles made of puffin beaks from North America—in the desert and on the ice we want to make music. There is a lamp made from a recycled light bulb from Chitungwiza township in Zimbabwe. But the most moving object in the whole museum is in a case acquired from the Masindi District refugee camp in Uganda in 1998: an empty insecticide tin, reshaped with wires and a plastic nozzle to make a toy, an elegant yellow aeroplane. Even in the most wretched circumstances, we dream of flight, and we share that dream with our children.

Pitt Rivers Museum, Oxford open daily; prm.ox.ac.uk