Science is one of humanity's greatest cultural achievements. By testing ideas against evidence, our species has been able to trace the course of evolution, to devise medicines that prolong lives, and even to glimpse the first microseconds of the universe. For all its success, though, there remains no shortage of people who doubt science's insights and achievements, sometimes preferring bizarre beliefs that count as unproven or disproved under its lens. Even the most compelling evidence cannot always dent the popularity of attractive but groundless stories.

This human reluctance to spurn a tall tale that we would like to be true is worth understanding, and the project on which Will Storr embarks in The Heretics: Adventures with the Enemies of Science is therefore an admirable one. In seeking out creationists and homeopaths, Holocaust-deniers and past-life regression therapists, he seeks not to mock strange convictions, but to get inside the minds of those who hold them.

The result is an entertaining journey dotted with some fascinating reportage. Storr tells good stories, which sometimes shed interesting light on the psychology and delusion of belief. His account of a tourist trip around concentration camps, with David Irving as a grumpy and self-absorbed guide, is outstanding, exposing both the historian's staggering ability to deceive himself, and a surprising diversity of behaviour among his neo-Nazi acolytes.

Other strong chapters detail morgellons – an itching syndrome that sufferers claim is caused by mysterious fibres, but which medicine does not recognise – and a damaged woman who disowned her family after "recovering" memories of Satanic abuse. In each case, Storr remains sufficiently dubious of improbable claims, but sufficiently open to the genuine distress of those who make them, to explore alternative explanations for their plight that fall the right side of Occam's razor.

It is the very quality of these chapters, however, that makes Storr's book disappointing and infuriating. For having shown he can achieve the kind of sceptical distance that makes Jon Ronson and Louis Theroux such effective chroniclers of weirdness, he casts it aside in an indulgence of wishful thinking. It is not enough for Storr to consider why people believe weird things; he also wants to challenge whether these things really are weird. He seems to accept, deep down, that they are, but he doesn't want to admit this. He is like the child who still wants to believe in Father Christmas, but who is just old enough to know better. Life would be more magical, more fun, if the story were true.

So it is that homeopaths are given a more sympathetic hearing than sceptics, with no discussion of the harm that unscientific medicine can do: Storr's storytelling does not extend to children like Neon Roberts, denied timely radiotherapy for his brain tumour as his mother sought an "alternative". Rupert Sheldrake, who posits that dogs can sense when their owners are coming home, shakes Storr's certainty that telepathy does not exist. "A new grey space has been nudged between the black and the white," he says. "And it is invigorating to have some mystery back."

For a book about the "enemies of science", Storr does remarkably little to engage with what science is. He explores the psychology and neuroscience of belief, explaining how all of us are prone to confirmation bias and cognitive dissonance: scientists are people, who fall into these traps from time to time. But he never considers that the great strength of the scientific approach is its recognition of these weaknesses of human thinking, and its attempts, albeit partial and imperfect, to erect defences against them. This is why randomised controlled trials, statistical significance and fair experimental design matter. They help us to avoid seeing only that which we find invigorating.

Also absent from The Heretics are the ideas of Thomas Bayes, the great statistician celebrated in Nate Silver's The Signal and the Noise. Bayes's insight was that a claim should be judged not only against new information about it, but also according to the prior probability of its truth. An observation that appears to contradict a well-attested phenomenon – such as faster-than-light neutrinos – is more likely to be noise than signal. To win support for hypotheses that are scientifically implausible – such as homeopathy, telepathy and morgellons – exceptional evidence is therefore needed. The onus is on the believer to find it, not on the sceptic to show it does not exist.

Without Bayes, Storr wastes time and space on trivialities that don't really matter. While his skill as an interviewer successfully paints James Randi, the sceptic champion, as an unpleasant bully who can be creative with the truth, he fails to recognise that this does not refute his arguments, nor confirm those of the heretics Randi debunks. "Bring on the psychics, bring on the alien abductees," says Storr. Yet even if Randi exaggerates from time to time, it remains far more probable that he, not the psychics, is right.

It is all very well to approach weird beliefs with an open mind, but as the aphorism goes, you must be careful not to leave it so open that your brains fall out.

Mark Henderson is author of The Geek Manifesto: Why Science Matters