Nick Brown does not look like your average student. He's 53 for a start and at 6ft 4in with a bushy moustache and an expression that jackknifes between sceptical and alarmed, he is reminiscent of a mid-period John Cleese. He can even sound a bit like the great comedian when he embarks on an extended sardonic riff, which he is prone to do if the subject rouses his intellectual suspicion.

A couple of years ago that suspicion began to grow while he sat in a lecture at the University of East London, where he was taking a postgraduate course in applied positive psychology. There was a slide showing a butterfly graph – the branch of mathematical modelling most often associated with chaos theory. On the graph was a tipping point that claimed to identify the precise emotional co-ordinates that divide those people who "flourish" from those who "languish".

According to the graph, it all came down to a specific ratio of positive emotions to negative emotions. If your ratio was greater than 2.9013 positive emotions to 1 negative emotion you were flourishing in life. If your ratio was less than that number you were languishing.

It was as simple as that. The mysteries of love, happiness, fulfilment, success, disappointment, heartache, failure, experience, random luck, environment, culture, gender, genes, and all the other myriad ingredients that make up a human life could be reduced to the figure of 2.9013.

It seemed incredible to Brown, as though it had been made up. But the number was no invention. Instead it was the product of research that had been published, after peer review, in no less authoritative a journal than American Psychologist – the pre-eminent publication in the world of psychology that is delivered to every member of the American Psychological Association. Co-authored by Barbara Fredrickson and Marcial Losada and entitled Positive Affect and the Complex Dynamics of Human Flourishing, the paper was subsequently cited more than 350 times in other academic journals. And aside from one partially critical paper, no one had seriously questioned its validity.

Masters student Nick Brown: 'If you want to be a whistleblower you have to be prepared to lose your job. I’m able to do what I’m doing here because I’m nobody.'

Fredrickson is a distinguished psychologist, a professor at the University of North Carolina, a winner of several notable psychology awards and bestselling author of a number of psychology books, including Positivity, which took her and Losada's academic research and recast it for a mass audience – the subtitle ran "Top-Notch Research Reveals the 3-to-1 Ratio That Will Change Your Life".

"Just as zero degrees celsius is a special number in thermodynamics," wrote Fredrickson in Positivity, "the 3-to-1 positivity ratio may well be a magic number in human psychology."

Fredrickson is the object of widespread admiration in the field of psychology. Martin Seligman, former president of the American Psychological Association and a bestselling author in his own right, went so far as to call her "the genius of the positive psychology movement". On top of which she is also an associate editor at American Psychologist.

By contrast, Brown was a first-term, first-year, part-time masters student who was about to take early retirement from what he calls a "large international organisation" in Strasbourg, where he had been head of IT network operations. Who was he to doubt the work of a leading professional which had been accepted by the psychological elite? What gave him the right to suggest that the emperor had gone naturist?

"The answer," says Brown when I meet him in a north London cafe, "is because that's how it always happens. Look at whistleblower culture. If you want to be a whistleblower you have to be prepared to lose your job. I'm able to do what I'm doing here because I'm nobody. I don't have to keep any academics happy. I don't have to think about the possible consequences of my actions for people I might admire personally who may have based their work on this and they end up looking silly. There are 160,000 psychologists in America and they've got mortgages. I've got the necessary degree of total independence."

Armed with that independence, he went away and looked at the maths that underpinned Fredrickson and Losada's ratio. Complex or non-linear dynamics are not easy for an untrained mathematician to understand, much less work out. Losada, who claimed expertise in non-linear dynamics, was working as a business consultant and making mathematical models of business team behaviour when he first met Fredrickson.

In Positivity, Fredrickson describes the moment when Losada explained how he could apply complex dynamics to her theories of positive psychology. "Hours into our lively discussion, he made a bold claim: based on his mathematical work, he could locate the exact positivity ratio that would distinguish those who flourished from those who didn't."

So impressed was she by this boast that Fredrickson arranged a sabbatical from her teaching duties "so I could immerse myself in the science of dynamic systems that Marcial had introduced me to".

There were several psychologists, versed in non-linear dynamics, who smelt something fishy about the maths in the published paper. Stephen Guastello, from Marquette University, wrote a note of mild complaint to American Psychologist, which it chose not to publish because "there wasn't enough interest in the article". Guastello feels now that he should have been more forceful in his opinions. "In retrospect," he says, "I see how I could have been more clearly negative and less supportive of what looked like an article that could move the field forward if someone would follow up with some strong empirical work."

John Gottman, a leading authority in the psychology of successful relationships, wrote to Losada because he couldn't follow the equations. "I thought it was something I didn't know about, because he's a smart guy, Losada. He never answered my email," he says. Gottman also wrote to Fredrickson. "She said she didn't understand the math either."

"Not many psychologists are very good at maths," says Brown. "Not many psychologists are even good at the maths and statistics you have to do as a psychologist. Typically you'll have a couple of people in the department who understand it. Most psychologists are not capable of organising a quantitative study. A lot of people can get a PhD in psychology without having those things at their fingertips. And that's the stuff you're meant to know. Losada's maths were of the kind you're not meant to encounter in psychology. The maths you need to understand the Losada system is hard but the maths you need to understand that this cannot possibly be true is relatively straightforward."

Brown had studied maths to A-level and then took a degree in engineering and computer science at Cambridge. "But I actually gave up the engineering because the maths was too hard," he says, laughing at the irony. "So I'm really not that good at maths. I can read simple calculus but I can't solve differential equations. But then neither could Losada!"

He went back over Losada's equations and he noticed that if he put in the numbers Fredrickson and Losada had then you could arrive at the appropriate figures. But he realised that it only worked on its own terms. "When you look at the equation, it doesn't contain any data. It's completely self-referential."

Unfortunately, while his grasp of maths was strong enough to see the problem, it wasn't sufficiently firm to be able to mount an academic takedown of Fredrickson's and Losada's work. Yet that was what he wanted to do. Once he knew to his own satisfaction that their research was fundamentally flawed, he was not going to be content to let things pass. So he decided to seek the help of an academic mathematician. Not just any academic mathematician either, but one who had made a name for himself by puncturing the bogus use of maths and science in another discipline.

Back in 1996, Alan Sokal wrote a paper called Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity and submitted it to an academic cultural studies journal called Social Text, which promptly published the article. As the title suggested, the paper was dense with impenetrable theory. Among other things, it disparaged the scientific method and western intellectual hegemony and claimed that quantum gravity could only be understood through its political context.

The paper, as Sokal quickly admitted, was a hoax, a deliberate pastiche of the sorts of nonsensical postmodern appropriations of maths and physics at which French critical theorists particularly excelled – among them Jacques Derrida, Jacques Lacan, Gilles Deleuze and Julia Kristeva. A major intellectual controversy ensued in which postmodernists stood accused of pseudo-science, absurd cultural relativism and the concealing of ignorance and innumeracy behind obscurantist prose. In response Sokal was derided as a pedant, a literalist and a cultural imperialist.

Despite the counterattacks, Sokal gained a reputation as a formidable enemy of bad science. As such he was regularly approached by people who believed they had uncovered an intellectual imposture, be it in architecture, history or musicology.

"I don't think I'm a crank," Brown had said in his email to Sokal. "I am just this grad student with no qualifications or credentials, starting out in the field. I don't know how to express this kind of idea especially coherently in academic written form, and I suspect that even if I did, it would be unlikely to be published."

But like many such requests, it began to disappear beneath a pile of other emails. It was only several weeks later that Sokal came across it again and realised that on this occasion he could help because it was in a field he knew something about: mathematics and physics.

Losada had derived his mathematical model from a system of differential equations known as the Lorenz equations, after Edward Lorenz, a pioneer of chaos theory.

"The Lorenz equation Losada used was from fluid dynamics," says Sokal, "which is not the field that I'm specialised in, but it's elementary enough that any mathematician or physicist knows enough. In 10 seconds I could see it was total bullshit. Nick had written a very long critique and basically it was absolutely right. There were some points where he didn't quite get the math right but essentially Nick had seen everything that was wrong with the Losada and Fredrickson paper."

Sokal did a little research and was amazed at the standing the Fredrickson and Losada paper enjoyed. "I don't know what the figures are in psychology but I know that in physics having 350 citations is a big deal," he says. "Look on Google you get something like 27,000 hits. This theory is not just big in academia, there's a whole industry of coaching and it intersects with business and business schools. There's a lot of money in it."

The concept of positive thinking dates back at least as far as the ancient Greeks. Throughout written history, metaphysicians have grappled with questions of happiness and free will. The second-century Stoic sage Epictetus argued that "Your will needn't be affected by an incident unless you let it". In other words, we can be masters and not victims of fate because what we believe our capability to be determines the strength of that capability.

In one way or another, positive thinking has always been concerned with optimising human potential, which is a key component of psychology. But in the 20th century, confronting the great traumas of two annihilating wars, the psychology profession became increasingly focused on the dysfunctional and pathological aspects of the human mind. The emphasis was on healing the ill rather than improving the well.

So it was left to popular or amateur psychology, and in particular that sector specialising in business success, to accentuate the positive. Books such as Norman Vincent Peale's The Power of Positive Thinking, published in 1952, became huge bestsellers. By the 1970s and 1980s, self-help had mushroomed into a vast literary genre that encompassed everything from the secrets of material achievement to the new age promises of chakras, reiki and self-realisation.

On becoming president of the American Psychological Association in 1998, Martin Seligman set out to bring scientific rigour to the issue of self-improvement. In his inaugural speech, he announced a shift in psychology towards a "new science of human strengths".

Barbara Frederickson, associate editor of American Psychologist, accepts the errors in the maths that Nick Brown pointed out, but still stands by her theory of positivity.

"It's my belief," said Seligman, "that since the end of the second world war, psychology has moved too far away from its original roots, which were to make the lives of all people more fulfilling and productive, and too much toward the important, but not all-important, area of curing mental illness."

He called for "a reoriented science that emphasises the understanding and building of the most positive qualities of an individual". It was an optimistic period in American history. The economy was buoyant, US geopolitical power was unchallenged and no major conflicts were raging. As a result, there was almost a messianic note of global ambition in Seligman's address. "We can show the world what actions lead to wellbeing, to positive individuals, to flourishing communities, and to a just society," he declared.

Suddenly a plethora of positive psychology books began to appear, written by eminent psychologists. There was Flow: The Psychology of Happiness by Mihaly Csikszentmihalyi, who with Seligman is seen as the co-founder of the modern positive psychology movement; Authentic Happiness: Using the New Positive Psychology to Realise Your Potential for Lasting Fulfilment by Seligman himself. And of course Fredrickson's Positivity, approved by both Seligman and Csikszentmihalyi. Each of them appeared to quote and promote one another, creating a virtuous circle of recommendation.

And these books were not only marketed like a previous generation of self-help manuals, they often shared the same style of cod-sagacious prose. "Positivity opens your mind naturally, like the water lily that opens with sunlight," writes Fredrickson in Positivity.

Then there was the lucrative lecture circuit. Both Seligman and Fredrickson are hired speakers. One website lists Seligman's booking fee at between $30,000 and $50,000 an engagement. In this new science of happiness, it seemed that all the leading proponents were happy.

But then Nick Brown started to ask questions.

Around the time Brown first came across Fredrickson's work, a case came to light in Holland in which a psychologist called Diederik Stapel, who was dean of faculty at Tilburg University, was caught by his graduate students making up data. It turned out he'd been falsifying his research for the previous 15 years. Brown, who is currently translating Stapel's autobiography, got in touch with him and asked him why he did it.

"The way he describes it," says Brown, "is that the environment was conducive to it. He said, 'I could either do the hard work or put my hand in the jar and take out a biscuit'." It does a massive amount of harm to science when this sort of thing happens. Nobody's accusing Fredrickson of making anything up. She just basically invented her own method. Is that worse than inventing your own data?"

After he had established contact with Sokal, Brown sent him a 15,000-word draft, which was much too long for publication. At first the professor agreed to give Brown advice on cleaning up the draft. He also told him that he should go to American Psychologist, and he contributed a pedagogic section, explaining the maths.

"I still wasn't thinking that I was going to be a co-author but Nick sent me drafts and I just liked his writing style," recalls Sokal. "It made me laugh. He had this gift for English understatement."

Getting their critique of Fredrickson into the publication of which she was an associate editor was a tall order. To help him get across the line, Brown had already recruited Harris Friedman, a sympathetic psychologist who had doubts about Fredrickson's claims but was not sufficiently versed in maths to make a case on his own.

Sending revised versions back and forth among themselves, the three men gradually composed what they considered to be a watertight argument. The initial title they submitted to American Psychologist was The Complex Dynamics of an Intellectual Imposture – an ironic play on Fredrickson and Losada's original piece. That was rejected by the editor because he argued that the word "imposture" implied a deliberate fraud on the part of Fredrickson and Losada.

Sokal insists that this was never their intention. As Brown puts it in characteristic manner. "This particular paper wasn't an act of fraud and it wasn't about statistics. It's that someone had a brain-fart one day."

Following much negotiation, Brown, Sokal and Friedman had their paper accepted by American Psychologist and it was published online last July under the only slightly less provocative title of The Complex Dynamics of Wishful Thinking. Referring to the bizarrely precise tipping point ratio of 2.9013 that Fredrickson and Losada trumpeted applied to all humans regardless of age, gender, race or culture, the authors – in fact Brown, in this sentence – wrote: "The idea that any aspect of human behaviour or experience should be universally and reproducibly constant to five significant digits would, if proven, constitute a unique moment in the history of the social sciences."

The paper mounted a devastating case against the maths employed by Fredrickson and Losada, who were offered the chance to respond in the same online issue of American Psychologist. Losada declined and has thus far failed to defend his input in any public forum. But Fredrickson did write a reply, which, putting a positive spin on things, she titled Updated Thinking on Positivity Ratios.

She effectively accepted that Losada's maths was wrong and admitted that she never really understood it anyway. But she refused to accept that the rest of the research was flawed. Indeed she claimed that, if anything, the empirical evidence was even stronger in support of her case. Fredrickson subsequently removed the critical chapter that outlines Losada's input from further editions of Positivity. She has avoided speaking to much of the press but in an email exchange with me, she maintained that "on empirical grounds, yes, tipping points are highly probable" in relation to positive emotions and flourishing.

"She's kind of hoping the Cheshire cat has disappeared but the grin is still there," says Brown, who is dismissive of Fredrickson's efforts at damage limitation. "She's trying to throw Losada over the side without admitting that she got conned. All she can really show is that higher numbers are better than lower ones. What you do in science is you make a statement of what you think will happen and then run the experiment and see if it matches it. What you don't do is pick up a bunch of data and start reading tea leaves. Because you can always find something. If you don't have much data you shouldn't go round theorising. Something orange is going to happen to you today, says the astrology chart. Sure enough, you'll notice if an orange bicycle goes by you."

But social psychology is full of theorising and much of it goes unquestioned. This is particularly the case when the research involves, as it does with Fredrickson, self-report, where the subjects assess themselves.

As John Gottman says: "Self-report data is easier to obtain, so a lot of social psychologists have formed an implicit society where they won't challenge one another. It's a collusion that makes it easier to publish research and not look at observational data or more objective data."

In general, says Gottman, the results of self-report have been quite reliable in the area of wellbeing. The problem is that when it comes down to distinguishing, say, those who "languish" from those who "flourish", there may be all manner of cultural and personal reasons why an individual or group might wish to deny negative feelings or even downplay positive ones.

"It's a lot more complicated than Fredrickson is suggesting," says Gottman.

After initially being turned down, Brown, Sokal and Friedman went through American Psychologist's lengthy appeals procedure and won the right to reply to Fredrickson's reply. They are currently working on what is certain to be a very carefully considered response. But it doesn't take a psychologist to work out that, given the nature of human behaviour, it's unlikely to be the last word.