Have you ever noticed that more people come back from heaven than from hell? We have all read those astonishing reports of near-death experiences (NDEs, as the aficionados call them) - the things that people say have happened to them when they almost, but don't quite, shuffle off the coil. They are nearly always pleasant and deeply reassuring in a saccharin-soaked way. Lots of spinning down warm, dark tunnels to the sound of celestial music; lots of trips along country lanes lined with hedges, towards the light of a welcoming cottage at the end of the road; lots of tumbling down Alice-in-Wonderland rabbit holes, but without the damaging effects of gravity.

True, Maurice Rawlings jnr, MD, heart surgeon in Chattanooga, Tennessee, and author of To Hell and Back, did have patients who reported very nasty NDEs after they came back on his operating table. Booming noises; licking flames and all that Mephistophelian stuff. But perhaps that tells us more about the challenges of living in Chattanooga, Tennessee, than about the metaphysics of life after death.

Predictably, the amazingly consistent, remarkably heaven-like experiences recounted by the majority of NDE-ers (yes, that really is what the experts call them) have been summarily dismissed by materialist sceptics - like me. Of course the brain does funny things when it's running out of oxygen.

The odd perceptions are just the consequences of confused activity in the temporal lobes. But NDEs have taken on a new cloak of respectability with a book by a Harvard doctor. Proof of Heaven, by Eben Alexander, will make your toes wiggle or curl, depending on your prejudices. What's special about his account of being dead is that he's a neurosurgeon. At least that's what the publicity is telling us.

It's a cover story in Newsweek magazine, with a screaming headline: ''Heaven is real: a doctor's account of the afterlife''. Just as you'd expect from a doctor, his account is precise and detailed. In the autumn of 2008, he contracted a very rare bacterial meningitis that he says made his brain ''shut down'' and put his ''higher-order brain functions totally off line''. The soup-like state of Dr Alexander's brain was, he writes, ''documented by CT scans'' (although CT scans don't say anything about the activity of the brain) and ''neurological examinations''.

Although the neurons of his cortex were ''stunned to complete inactivity by the bacteria'', his conscious self journeyed into another world. There was wonderful music and light. There were clouds, ''big, puffy, pink-white ones that showed up sharply against the deep blue-black sky''. And there were angels (well, perhaps birds): ''flocks of transparent, shimmering beings''. But then it gets really weird. It turns out that he wasn't alone. ''For most of my journey, someone was with me. A woman.'' She had a lovely face and golden brown tresses, and she was dressed appropriately for a Cecil B DeMille movie, in peasant costume. She was quite a stunner. She looked at Dr Alexander ''with a look that, if you saw it for five seconds, would make your whole life up to that point worth living, no matter what had happened in it so far''. It was a look ''beyond all the different compartments of love we have down here on Earth''.

Well, many of us, after a couple of pints in the pub with our chums, might say that we've had that kind of experience; but not with a woman flying on a butterfly wing, as Dr Alexander's companion was. Although he ''still had little language function'' he was able to chat with the peasant lady, asking where he was and why he was there. He was overwhelmed by the answers, which ''came instantly in an explosion of light, colour, love and beauty that blew through me like a crashing wave''. After the clouds and the angels and the peasant lady, Dr Alexander went on to a ''pitch-black'' void, ''brimming with light'' from a ''brilliant orb'' that acted as an interpreter, explaining that the ''universe itself was like a giant cosmic womb''.

You may have sensed a subtle hint of scepticism in my account. As Eben Alexander says, he considers himself a faithful Christian, and it's therefore not surprising that he interpreted the chaos in his brain when he was almost dying in terms of his model of the afterlife. His, and the multitude of other memories reported by people who have been close to death, have to be seen first through the prism of hard science. The crucial question is not whether such astounding experiences should lead us to abandon materialist accounts of brain function, but whether materialist accounts can possibly explain them.

Peter Fenwick, senior lecturer at King's College, London, consultant at the Institute of Psychiatry, and president of the British branch of the International Association for Near Death Studies, acknowledges that there are deep problems in interpreting first-person memories of experiences that are supposed to have happened when the brain was out of action. Since the lucky survivor can only tell you about them after the event, how can we be sure that these things were perceived and felt at the time that their brains were messed up, rather than being invented afterwards? The same problem applies to dreams, indeed to any memory. Memory is notoriously fallible, and is treacherously easily misled by expectation. Cognitive psychologist Elizabeth Loftus has done brilliant experiments showing how the recall of real experiences can be transformed by what people think should have happened, and by what they are told might have happened.

In 150 years the science of perception has taught us that the way we appreciate the world around us is as much dependent on our expectations, our experiences, our inferences, as it is on the hard evidence of images on our retinas or vibrations in our ears. Remember when you have seen a face in the flickering flames of a fire, or saw a person in the distance as you walked along at night - only to discover that the face in the fire disappears with the next burst of flame and the person in the dark is just a letterbox. Is it not significant that the NDEs of Christians are full of biblical images? Either this confirms the correctness of their faith or it says that NDEs, like normal perception and memory, are redolent of culture, personal prejudice and past experience.

Perhaps if Eben Alexander were a Muslim, there would have been the mythical 72 virgins on the butterfly wing, rather than the bucolic one. If he were a Buddhist he would be called a delok, a person who has seemingly died, but who travels into bardo - an afterlife state - guided by a Buddhist deity. What Dr Alexander and his PR people claim is that his description of the afterlife is more authentic because he is a neurosurgeon. But when there is no evidence except the word of the beholder, a scientist's accounts are no more reliable than those of anyone else.

Would we literally believe the contents of a scientist's dream because he or she has a PhD?

Science has progressed by challenge and disagreement. But what is needed to consider seriously the kinds of claims made by Dr Alexander is not flowery prose and hyperbolic headlines. It's hard evidence. But I am trying (not very convincingly, I know) to keep an open mind. I remember the story of the nobleman who asked Zen Master Hakuin: ''What happens to the enlightened man at death?'' ''Why ask me?'' said Hakuin.

''Because you're a Zen master.''

''Yes, but not a dead one.''

Tales From That Bourn Whence Some Do Return