No one was more outraged than Helen Pidd when a South African scientist announced that dolphins are stupid. Would a day with Puck, Flo and Roxanne settle things one way or the other?

How many elderly goldfish do you know who could moonwalk on water to Kylie? This is the question with which I am wrestling as a 41-year-old bottlenose dolphin called Puck slides backwards across her 6m-deep pool, her body upright, tail smoothly making waves to the rhythm of Can't Get You Out Of My Head.

There is clearly something special about Puck and not just because in dolphin terms she ought really to have been pensioned off long ago. Yet according to controversial remarks made last month by a South African neuroscientist, Puck and her peers are less advanced than goldfish.

For anyone who grew up glued to Flipper on television ("No one you see /is smarter than he" ran the jaunty theme tune), it's a hard claim to swallow. But sometimes in life you have to accept that the dogmas you have long held dear - be it belief in Father Christmas, anti-ageing formulas, eternal love or anything else - might have been a great fat lie. So I have come to the dolphinarium at Boudewijn Seapark in Bruges to investigate. With the help of Puck, Roxanne, Flo, Yolta and Milo and the other supposedly dimwitted mammals, I am determined to find out the truth. (Why Belgium rather than Britain? Since 1993 it has been illegal to keep dolphins in captivity in this country.)

The trainers at Boudewijn, along with much of the dolphin-loving world, are deeply unimpressed with claims made by Paul Manger, a 40-year-old professor of neuroscience from the University of Witwatersrand in Johannesburg. In a weighty scientific paper published earlier this year in the Biological Reviews of the Cambridge Philosophical Society, Manger hypothesised that "there is no neural basis for the often-asserted high intellectual abilities of cetaceans." In other words, despite their supersized brains, cetaceans (whales, dolphins and porpoises), are profoundly thick.

This claim flew in the face of almost everything else published about the mammals recently and, indeed, ever. Which is why people are so touchy about it. "No one I have talked to in the scientific field takes the claims Manger makes in this paper seriously," snaps dolphin expert Lori Marino, senior lecturer in neuroscience and behavioural biology at Emory University, Atlanta.

The dolphin partly owes its reputation as a brainbox to something that scientists like to complicate by calling the encephalisation quotient, which says that the relative amount of brain per unit of body size can be used as a direct estimate of the intelligence of a species. In English, this means: big brain + not too big body = clever animal. Modern humans possess the highest level of encephalisation of mammals, as our brains are seven times the size you would expect for our body size, but dolphins are not far behind.

The earliest anecdote supporting cetaceans' supposed super-intelligence is probably the tale of Arion, the finest lyre-player of his day (c600 BC). When turfed off his ship by villains, this chap, so goes the tale, was rescued by a dolphin who was attracted by his singularly high-pitched singing.

But there is more recent evidence. In May, for example, researchers from St Andrews University reported that bottlenose dolphins adopt "signature whistles" to identify each other, just as humans use names. Another group studying dolphins in western Australia in 2005 noted that some of the dolphins used tools - bits of marine sponge foraged and attached to their snouts, to stop their noses scraping painfully against coral as they fished. Then there are the Irrawaddy dolphins in the Ayeyarwady river in Myanmar (formerly Burma), which help local fisherman by corralling fish into their nets.

Perhaps best of all, a group of researchers allegedly recently taught dolphins to "sing" the Batman theme tune. I was pretty sceptical about that one, until I came to Belgium. In one of the highlights of Boudewijn's twice daily live spectacle, a lucky girl is plucked from the audience and shown how to "conduct" the dolphins in a rather delightful round. She does this by wringing both hands as if opening jars, which seems to keep the dolphins time as they "sing" (OK, yelp) along.

This hand-wringing isn't the only sign language the mammals appear to understand. With the requisite flick of the trainers' wrists, they can open their mouths as if to laugh, nod their heads from side to side and (my personal favourite, along with the bit when they play football with their tails) blow air bubbles like smoke rings in the water. Manger would say that this is indicative of stimulus-response conditioning - ie, it can be taught by a good trainer and is not in any way a sign of a high-level of intelligent behaviour. Though this may be true - the dolphins are certainly encouraged by the promise of mackerel and pilchards from brightly coloured buckets, as well as high-pitched toots on the dog whistles worn round the trainers' necks - there is no denying one thing: it's really very, very cool.

But back to the new claims. On the off-chance you don't fancy wading through Manger's 46 pages, here is a dummies' synopsis of the key points. Dolphins have bigger than average brains - some weighing in excess of 8kg - but this has nothing to do with Stephen Hawkingesque neurological brilliance. On the contrary, the dolphin brain is not built for complex information processing, but rather is designed to counter the thermal challenges of being a warm-blooded mammal in a cold-water world. And finally, the comment that really riled the dolphin lobby: while there is plenty of anecdotal evidence to support the theory that dolphins are exceptionally intelligent, no one has ever conclusively proved that they are.

"You put an animal in a box, even a lab rat or gerbil, and the first thing it wants to do is climb out of it," said Manger in interviews to publicise his paper. "If you don't put a lid on top of the bowl, a goldfish will eventually jump out to enlarge the environment it is living in. But a dolphin will never do that."

I muse on this as I sit in the empty seats of the Belgian dolphinarium after the live spectacle, and two dolphins swim diligently in a tiny holding pool while their pals in the main tank practice double somersaults with the trainer. I know that the sidelined pair can jump the necessary distance: I saw them doing it in their show earlier. But, like obedient children, they only enter the main pool when their underwater trapdoors are opened.

So Manger's observation may be correct, but does this actually tell us anything useful? Not according to Lori Marino, senior lecturer in neuroscience and behavioural biology at Emory University. She has studied dolphins for 15 years and is one of Manger's biggest critics. "That study with the goldfish he is talking about was never published," she says. "It was presented at a meeting in 2000 and was never peer-reviewed, and I was there, and there are so many flaws in it I really don't know where to begin."

Marino claims that while it is true that dolphins have problems jumping over tuna nets, for example, this says nothing about their lack of intelligence. "It's just like if humans get trapped in a burning building, they panic. Dolphins, like humans, get stressed very easily - in fact, they can die of stress - and when they panic, they can't think straight."

But according to Manger, Marino and her doubting colleagues are too blinkered to even consider a brave new claim such as his. "People just don't want to believe it," says Manger in an interview with the Guardian. "It's a knee-jerk reaction. I think it's kinda cute that people are so upset about it."

Not everyone can see the cute side. Dolphins, with their happy, toothy "grins" and impressive repertoire of tricks and emotions, are universally loved - some 77,000 people apparently belong to the Whale and Dolphin Conservation Society worldwide - and slagging them off is heresy. Manger has even been accused of leading dolphins to an early grave. One South African amateur dolphin expert, Nan Rice, who has worked for 35 years with the Dolphin Action and Protection Group in Cape Town, says Manger's research could lead to the animals' exploitation - and possible extinction. "If you tell people that dolphins are not intelligent, they will start exploiting them and at the end we won't have many of them left."

Leaving aside whether Manger's comments are dangerous or not, one big problem when discussing dolphin neurology is the fact that, because of laws protecting marine mammals, no invasive experiments of the cetacean brain have been carried out. Unlike, say rats, you can't simply catch yourself a dolphin and dig around in their brain and generally experiment on them while they are still alive. Similarly, for reasons of practicality as well as ethics, the vast majority of dolphin experiments have been conducted on dolphins born in captivity. And who knows how representative these animals, which have never seen open water, can really be of the species as a whole.

The other major stumbling block is defining what we mean by intelligence. There is no reliable, universally accepted method to measure it in humans. A Mensa test? A-levels? Those multiple-choice quizzes in women's magazines?

Nevertheless, contrary to Manger's claim, there have actually been a great number of experiments that seem to indicate that dolphins aren't just pretty creatures. One of the most interesting regards self-recognition and comes from Diana Reiss of Colombia University, along with Marino. In a 2001 paper, they showed that dolphins can recognise themselves in the mirror - a skill previously thought to be unique to humans and apes.

The scientists exposed two bottlenose dolphins to reflective surfaces after marking the dolphins with black ink, applying a water-filled marker (sham-marking) or not marking them at all. The team predicted that if the dolphins - which had prior experience with mirrors - recognised their reflections, they would not show social responses; they would spend more time in front of the mirror when marked; and they would make their way over to the mirror more quickly to inspect themselves when marked or sham-marked. The experiments bore out all three predictions in both dolphin subjects. Moreover, the animals even selected the best reflective surface available to view their markings. "I've seen it done," says Johan Cottyn, head trainer at Boudewijn. His colleague, vet Piet de Laender, nods enthusiastically. "Dolphins realise that they are looking at themselves. It doesn't work with dogs. Say if you put a spot on a dog's head and put them in front of a mirror, they will think oh, there's a dog with a spot on its head, I'll go and say hello. A dolphin would stay and inspect its new look." This doesn't just tell us that dolphins are vain: it shows that they are aware of themselves as individuals, which takes a high level of cognitive skill.

As well as pointing out dolphins' ability to learn fast and understand complicated language-like commands (which is also true of the great apes), Cottyn is most impressed by the way he and his team have been able to train their dolphins to accept medical interventions without a fuss. "One of them had a kidney problem, which meant she had to have an injection daily. Even though it really hurt her, because the needle had to cut through the layer of blubber, she seemed to understand that in order to survive, she had to take the jab, and would automatically come for it every day."

Cottyn talks of how the Boudewijn dolphins can communicate to humans beyond what they have been explicitly taught. "Once, when one of the dolphins was giving birth the baby got stuck halfway out. We came to see what was going on and the mother rubbed her tummy to show us where the problem was," he says.

Numerous studies have also shed light on dolphins' rationality, high degree of sociability, social awareness and cognitive functioning.

But de Laender has no time for this whole debate, and especially not for anyone who tries to anthropomorphise dolphins. "They're not humans, and trying to judge them by human standards of intelligence is pointless," he says. "So at the end of the day, it's a waste of time trying to say definitively whether dolphins are clever or not, and anyone who tells you they have the answer is bullshitting".